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BY 4.0 license Open Access Published by De Gruyter Open Access December 31, 2020

Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory

Ayman Badawi and Roswitha Rissner
From the journal Open Mathematics

Abstract

For a partially ordered set ( A , ) , let G A be the simple, undirected graph with vertex set A such that two vertices a b A are adjacent if either a b or b a . We call G A the partial order graph or comparability graph of A. Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G = G A . For a class C of simple, undirected graphs and n, m 1 , we define the Ramsey number C ( n , m ) with respect to C to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in C consisting of r vertices contains either a complete n-clique K n or an independent set consisting of m vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.

MSC 2010: 06A06; 05CXX; 05D10

1 Introduction

The Ramsey number ( n , m ) gives the solution to the party problem, which asks for the minimum number ( n , m ) of guests that must be invited so that at least n will know each other or at least m will not know each other. In the language of graph theory, the Ramsey number is the minimum number v = ( n , m ) of vertices such that all undirected simple graphs of order v contain a clique of order n or an independent set of order m. There exists a considerable amount of literature on Ramsey numbers. For example, Greenwood and Gleason [1] showed that ( 3 , 3 ) = 6 , ( 3 , 4 ) = 9 and ( 3 , 5 ) = 14 ; Graver and Yackel [2] proved that ( 3 , 6 ) = 18 ; Kalbfleisch [3] computed that ( 3 , 7 ) = 23 ; McKay and Min [4] showed that ( 3 , 8 ) = 28 and Grinstead and Roberts [5] determined that ( 3 , 9 ) = 36 .

A summary of known results up to 1983 for ( n , m ) is given in the study by Chung and Grinstead [6]. An up-to-date-list of the best currently known bounds for generalized Ramsey numbers (multicolor graph numbers), hypergraph Ramsey numbers and many other types of Ramsey numbers is maintained by Radziszowski [7].

In this paper, we determine the Ramsey number of partial order graphs. We want to point out that recently, a colleague kindly made us aware that such graphs in the literature are also known as comparability graph and our result Theorem 2.2 is a consequence of [8, Theorem 6] (also see [8, Corollary 1]). However, our proof of Theorem 2.2 is self-contained and it is completely different from the proof in [8]. Our proof solely relies on the pigeonhole principle. For a partially ordered set ( A , ) , let G A be the simple, undirected graph with vertex set A such that two vertices a b A are adjacent if either a b or b a . We call G A the partial order graph (comparability graph) of A. In this paper, we will just use the name partial order graph. Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G = G A . For a class C of simple, undirected graphs and n, m 1 , we define the Ramsey number C ( n , m ) with respect to the class C to be the minimal number r of vertices such that every induced subgraph of an arbitrary graph in C consisting of r vertices contains either a complete n-clique K n or an independent set consisting of m vertices.

Next, we remind the readers of the graph theoretic definitions that are used in this paper. We say that a graph G is connected if there is a path between any two distinct vertices of G. For vertices x and y of G, we define d ( x , y ) to be the length of a shortest path from x to y ( d ( x , x ) = 0 and d ( x , y ) = if there is no such path). The diameter of G is diam ( G ) = sup { d ( x , y ) | x and y are vertices of G } . The girth of G, denoted by g ( G ) , is the length of a shortest cycle in G ( g ( G ) = if G contains no cycles). We denote the complete graph on n vertices or n-clique by K n and the complete bipartite graph on m and n vertices by K m , n . The clique number ω ( G ) of G is the largest positive integer m such that K m is an induced subgraph of G. The chromatic number of G, χ ( G ) , is the minimum number of colors needed to produce a proper coloring of G (that is, no two vertices that share an edge have the same color). The domination number of G, γ ( G ) , is the minimum size of a set S of vertices of G such that each vertex in G \ S is connected by an edge to at least one vertex in S by an edge. An independent vertex set of G is a subset of the vertices such that no two vertices in the subset are connected by an edge of G. For a general reference for graph theory we refer to Bollobás’ textbook [9].

In Section 2, we show that the Ramsey number P o G ( n , m ) for the class P o G of partial order graphs equals ( n 1 ) ( m 1 ) + 1 , see Theorem 2.2. In Section 3, we study subclasses of partial order graphs that appear in the context of ring theory. Among other results, we show that for the classes P D G of perfect divisor graphs, D i v G of divisibility graphs, I n G of inclusion ideal graphs, a t G of matrix graphs and Idem G of idempotent graphs of rings, the respective Ramsey numbers equal to P o G , see Theorems 3.4, 3.8, 3.12, 3.16 and 3.21, respectively. In Section 4, we a present a subclass of partial ordered graphs with respect to which the Ramsey numbers are non-symmetric.

Throughout this paper, Z and Z n will denote the integers and integer modulo n, respectively. Moreover, for a ring R we assume that 1 0 holds, R = R \ { 0 } denotes the set of non-zero elements of R and U ( R ) denotes the group of units of R.

2 Ramsey numbers of partial order graphs

Definition 2.1

  1. For a partially ordered set ( A , ) , let G A be the simple, undirected graph with vertex set A such that two vertices a b A are adjacent if either a b or b a . We call G A the partial order graph of A. Furthermore, we say that G is a partial order graph if there exists a partially ordered set A such that G = G A . By P o G we denote the class of all partial order graphs.

  2. For a class C of simple, undirected graphs and n, m 1 , we set C ( n , m ) to be the minimal number r of vertices such that every induced subgraph of an arbitrary graph in C consisting of r vertices contains either a complete n-clique K n or an independent set consisting of m vertices. We call C the Ramsey number with respect to the class C .

Theorem 2.2

Let n, m 1 (n, m need not be distinct). Then for the Ramsey number P o G with respect to the class P o G of partial order graphs, the following equality holds

P o G ( n , m ) = P o G ( m , n ) = ( n 1 ) ( m 1 ) + 1 .

Proof

First, we prove that P o G ( n , m ) > ( n 1 ) ( m 1 ) . Let A be a set of cardinality ( n 1 ) ( m 1 ) and A 1 , , A n 1 an arbitrary partition of A into n 1 subsets each of cardinality m 1 . Furthermore, for a, b A , we say a b if and only if a = b or a A i and b A j with i < j . Then is a partial order on A and the partial order graph G A is a complete ( n 1 ) -partite graph in which each partition has m 1 independent vertices. It is easily verified that the clique number of G A is n 1 and that at most m 1 vertices of G A are independent.

Let G be a partial order graph and H an induced subgraph. We show that if H contains ( n 1 ) ( m 1 ) + 1 vertices, then H contains either an n-clique K n or an independent set of m vertices.

Let G dir be the directed graph with the same vertex set as G such that ( a , b ) is an edge if a b and a b . Then H dir (the subgraph of G dir induced by the vertices of H) contains a directed path of length n if and only if H contains an ( n + 1 ) -clique K n + 1 .

Note that G dir does not contain a directed cycle. This allows us to define pos H ( a ) to be the maximal length of a directed path in H dir with endpoint a for a vertex a of H.

It is easily seen that pos H ( b ) pos H ( a ) 1 for every edge ( b , a ) in H dir . In particular, if for two vertices a, b of H, pos H ( a ) = pos H ( b ) , then the two vertices are independent in H.

Moreover, a straight-forward argument shows that H contains an n-clique K n if and only if there exists a vertex a in H with pos H ( a ) n 1 .

Now, assume that H does not contain an n-clique K n . This implies that pos H ( a ) < n 1 for all vertices a in H. It then follows by the pigeonhole principle that among the ( n 1 ) ( m 1 ) + 1 vertices in H, there are at least m vertices a with pos H ( a ) = k for some k, 0 k n 2 . Therefore, H contains m independent vertices.

Since ( n 1 ) ( m 1 ) + 1 is symmetric in n and m, it further follows that P o G ( n , m ) = P o G ( m , n ) .□

3 Subclasses of partial order graphs that appear in the ring theory

In this section, we discuss subclasses of partial order graphs that appear in the context of ring theory. In particular, we focus on the implications of Theorem 2.2. Recall for a class C of graphs, C denotes the Ramsey number with respect to C , cf. Definition 2.1.

3.1 Perfect divisor graphs

Definition 3.1

Let R be a commutative ring, n 2 and S = { m 1 , , m n } R \ U ( R ) be a set of n pairwise coprime non-zero non-units and m = m 1 m 2 m n . (Note that m = 0 is possible.)

  1. We say d is a perfect divisor of m with respect to S if d m and d is a product of distinct elements of S.

  2. The perfect divisor graph P D G ( S ) of S is defined as the simple, undirected graph ( V , E ) , where V = { d | d perfect-divisor of m } is the vertex set and for two vertices a b V , ( a , b ) E if and only if a | b or b | a .

  3. By P D G we denote the class of all perfect divisor graphs.

Lemma 3.2

Let R be a commutative ring, n 2 and S = { m 1 , , m n } R \ U ( R ) be a set of n pairwise coprime non-zero non-units and m = m 1 m 2 m n . Furthermore, let

V = { d | d p e r f e c t d i v i s o r o f m w i t h r e s p e c t t o S }

and define on V such that for all a, b V , we have a b if and only if a = b or a | b .

Then ( V , ) is a partially ordered set of cardinality | V | = 2 n 2 and PDG ( S ) is a partial order graph.

Proof

The relation clearly is reflexive and transitive, we prove that it is also antisymmetric. Let d V be a perfect divisor of m with respect to S. Then d = j J m j for J { 1 , , n } . We show that for every 1 i m , m i | d if and only if i J .

Obviously, if j J , then m i | d . Let us assume that i { 1 , , n } \ J . Then by hypothesis, for j J there are elements a j and b j R such that a j m j + b j m i = 1 holds. Hence,

1 = j J ( a j m j + b j m i ) = ( j J a j m j + c m i = a d + c m i

for some a, c R . Therefore, d and m i are coprime elements of R which in particular implies that m i d .

It follows that if d 1 and d 2 are distinct perfect divisors of m and d 1 | d 2 , then d 2 d 1 . Thus, ( V , ) is a partially ordered set.

Moreover, it follows that the elements in V correspond to the non-empty proper subset of { 1 , , n } . Therefore, their number amounts to

| V | = { J { 1 , , n } } = i = 1 n 1 n i = 2 n 2 .

Theorem 3.3

Let R be a commutative ring, n 2 and S = { m 1 , , m n } R \ U ( R ) be a set of n pairwise coprime non-zero non-units, m = m 1 m 2 m n and P D G ( S ) the perfect divisor graph of m with respect to S.

Then the following assertions hold:

  1. PDG ( S ) is a connected graph if and only if n 3 .

  2. If n 3 , then the diameter diam ( PDG ( S ) ) = 3 .

  3. The domination number of PDG ( S ) is equal to 2 if n 2 and equal to 1 if n = 1 .

  4. If n 3 , then the vertices in P k = { j J m j | | J | = k } for 1 k n 1 are pairwise not connected by an edge. In particular, PDG ( S ) is an ( n 1 ) -partite graph.

  5. If a P k = { j J m j | | J | = k } for 1 k n 1 , then deg ( a ) = 2 k + 2 n k 4 .

  6. If n 3 , then for the girth of PDG ( S ) the following holds

    g ( PDG ( S ) ) = 6 n = 3 , 3 n 4 .

  7. PDG ( S ) is planar if and only if n { 3 , 4 } .

Proof

(1) If n = 2 , then V consists of two vertices m 1 and m 2 which are coprime and hence not connected. Assume n 3 and let a = j J m j and b = k K m k be two distinct vertices of PDG ( S ) . Suppose that m j = m k for some j J and k K . Then a m j b is a path of length 2 from a to b if m j a , b and ( a , b ) is an edge otherwise. Suppose that m j m k for every j J and k K . We show that | J | n 2 or | K | n 2 . Suppose that | J | = | K | = n 1 . Since n 3 and m j m k for every j J and k K , we conclude that | { m j | j J } { m k | k K } | = 2 n 2 > n , a contradiction. Thus, | J | n 2 or | K | n 2 . Without loss of generality, we may assume that | J | n 2 . Take arbitrary k K . Then, a a m k m k b is a path of length 3 from a to b if b m k and otherwise, a a m k b is a path of length 2. Hence, PDG ( S ) is connected which completes the proof of (1).

(2) Suppose that n 3 . Then PDG ( S ) is connected by (1). Let a , b be two distinct vertices of PDG ( S ) . In light of the proof given in (1), we have d ( a , b ) 3 . Let a = j = 1 ( n 1 ) m j and b = m n . Then a m 1 m 1 b b is a shortest path in PDG ( S ) from a to b. Hence, d ( a , b ) = 3 . Thus, diam ( PDG ( S ) ) = 3 .

For (3) observe that every perfect divisor d of m is either divisible by m 1 or divides m 2 m 3 m n . Hence, every vertex of PDG ( S ) is connected by an edge to either one of these two vertices.

(4) Let 1 k n 1 and J, K { 1 , , n } with | J | = | K | = k . Set a = j J m j and b = k K m k be two different vertices of PDG ( S ) , which implies J K . Therefore, there exist j J \ K and k K \ J . In the proof of Lemma 3.2, we have shown that it now follows that m j b and m k a . In particular, it follows that a b and b a . Hence, no two vertices in { j J m j | | J | = k } are connected by an edge.

For (5), let a = j J m j be perfect divisor of m and set k = | J | . The perfect divisors of m with respect to S which divide a which are connected by an edge to a correspond to the non-empty, proper subsets of J which are i = 1 k 1 k i = 2 k 2 many. In addition, we need to count the number of perfect divisors of m which are divisible by a. These are exactly the ones of the form k K m k with J K { 1 , , n } of which there are i = 1 n k 1 n k i = 2 n k 2 . Hence, deg ( a ) = 2 k + 2 n k 4 .

(6) For n = 3 , we can verify in Figure 1 that there is cycle of length 6 and no shorter cycle.

If n 4 , then m 1 m 2 m 3 is a perfect divisor and the edges ( m 1 , m 1 m 2 ) , ( m 1 m 2 , m 1 m 2 m 3 ) and ( m 1 m 2 m 3 , m 1 ) form a cycle of length 3 which is the smallest possible length of a cycle in PDG ( S ) .

Finally, for (7), it is easily verified that PDG ( S ) is planar if n = 3 , cf. Figure 1. Moreover, Figure 2 shows a planar arrangement of the edges of PDG ( S ) for n = 4 .

If, however, n 5 , then Figure 3 is a K 3 , 3 subgraph of PDG ( S ) , and hence PDG ( S ) is not a planar by Kuratowski’s Theorem on planar graphs.□

Next, we compute the Ramsey number with respect to the class of perfect divisor graphs. Note that P D G is a subclass of P o G which immediately implies that P D G ( n , m ) P o G ( n , m ) for all n, m 1 . We use Theorem 3.3 to show that equality holds.

Figure 1 
                  Perfect divisor graph for 
                        
                           
                           
                              n
                              =
                              3
                           
                           n=3
                        
                     .

Figure 1

Perfect divisor graph for n = 3 .

Figure 2 
                  Perfect divisor graph for 
                        
                           
                           
                              n
                              =
                              4
                           
                           n=4
                        
                      with planar arrangement of edges.

Figure 2

Perfect divisor graph for n = 4 with planar arrangement of edges.

Figure 3 
                  
                     
                        
                           
                           
                              PDG
                              (
                              S
                              )
                           
                           \text{PDG}(S)
                        
                      contains 
                        
                           
                           
                              
                                 
                                    K
                                 
                                 
                                    3,3
                                 
                              
                           
                           {K}_{\mathrm{3,3}}
                        
                      as a subgraph for 
                        
                           
                           
                              n
                              ≥
                              5
                           
                           n\ge 5
                        
                     .

Figure 3

PDG ( S ) contains K 3,3 as a subgraph for n 5 .

Theorem 3.4

Let n, m 1 . Then for the Ramsey number P D G with respect to the class P D G of perfect divisor graphs the following holds:

P D G ( n , m ) = P o G ( n , m ) = ( n 1 ) ( m 1 ) + 1 .

Proof

We set w = ( n 1 ) ( m 1 ) and show that P D G ( n , m ) > w = ( n 1 ) ( m 1 ) by giving an example of perfect divisor graph G and an induced subgraph H of G with w vertices which is a complete ( n 1 ) -partite graph on w vertices in which independent sets are of cardinality at most m 1 .

Let R = Z and let S = { p 1 , p 2 , , p w } be a set of w distinct positive prime numbers of Z . We set m = p 1 p 2 p w and G = PDG ( S ) .

For each 1 i n 1 , let k i = ( i 1 ) ( m 1 ) and we set a i = p 1 p 2 p k i (where a 1 = 1 ) and

A i = { a i p k i + 1 , , a i p k i + ( m 1 ) } .

Note that A 1 = { p 1 , , p m 1 } .

Let H be the subgraph of G induced by the vertex set A 1 A 2 A n 1 . By construction, for each 1 i n 1 , | A i | = m 1 holds and A i is contained in the partition P k i + 1 of G, cf. Theorem 3.3.4. This implies that each A i is an independent vertex set of H of cardinality m 1 .

Moreover, since G is a ( w 1 ) -partite graph and each A i is contained in P k i + 1 , it follows that H is an ( n 1 ) -partite graph (with partitioning A 1 A 2 A n 1 ). For an example of this construction with m = 5 and n = 4 see Example 3.5.

Thus, not more than m 1 vertices of H are independent, and a straight-forward verification shows that the clique number of H is at most n 1 . Thus, P D G ( n , m ) > w . Hence, by Theorem 2.2, we have P D G ( n , m ) = P o G ( n , m ) = w + 1 = ( n 1 ) ( m 1 ) + 1 .□

Example 3.5

We demonstrate the construction of the previous proof for the example R = Z with n = 4 and m = 5 . That is, we construct a perfect divisor graph which has a complete 3-partite graph H as subgraph and each of the partitions of H consist of four independent vertices.

Let w = ( n 1 ) ( m 1 ) = 12 and we set S = { p 1 , p 2 , , p 12 } . Next, let n i = ( i 1 ) ( m 1 ) for 1 i 3 , that is, n 1 = 0 , n 2 = 4 and n 3 = 8 . Then a 1 = 1 , a 2 = p 1 p 2 p 3 p 4 and a 3 = p 1 p 2 p 8 .

We set

A 1 = { a 1 p 1 , a 1 p 2 , a 1 p 3 , a 1 p 4 } = { p 1 , p 2 , p 3 , p 4 } , A 2 = { a 2 p 5 , a 2 p 6 , a 2 p 7 , a 2 p 8 } = { ( p 1 p 4 ) p 5 , ( p 1 p 4 ) p 6 , ( p 1 p 4 ) p 7 , ( p 1 p 4 ) p 8 } , A 3 = { a 3 p 9 , a 3 p 10 , a 3 p 11 , a 3 p 12 } = { ( p 1 p 2 p 8 ) p 9 , ( p 1 p 2 p 8 ) p 10 , , ( p 1 p 2 p 8 ) p 12 } .

The subgraph of PDG ( S ) induced by A 1 A 2 A 3 is a complete 3-partite graph in which each partition has four vertices that are independent (Figure 4).

Figure 4 
                  Induced subgraph H of 
                        
                           
                           
                              PDG
                              (
                              {
                              
                                 
                                    p
                                 
                                 
                                    1
                                 
                              
                              ,
                              …
                              ,
                              
                                 
                                    p
                                 
                                 
                                    12
                                 
                              
                              }
                              )
                           
                           \text{PDG}(\{{p}_{1},\ldots ,{p}_{12}\})
                        
                      where, for better visibility, the edges between 
                        
                           
                           
                              
                                 
                                    A
                                 
                                 
                                    1
                                 
                              
                           
                           {A}_{1}
                        
                      and 
                        
                           
                           
                              
                                 
                                    A
                                 
                                 
                                    3
                                 
                              
                           
                           {A}_{3}
                        
                      are “hidden” behind the edges between 
                        
                           
                           
                              
                                 
                                    A
                                 
                                 
                                    1
                                 
                              
                           
                           {A}_{1}
                        
                      and 
                        
                           
                           
                              
                                 
                                    A
                                 
                                 
                                    2
                                 
                              
                           
                           {A}_{2}
                        
                      and the edges between 
                        
                           
                           
                              
                                 
                                    A
                                 
                                 
                                    2
                                 
                              
                           
                           {A}_{2}
                        
                      and 
                        
                           
                           
                              
                                 
                                    A
                                 
                                 
                                    3
                                 
                              
                           
                           {A}_{3}
                        
                     .

Figure 4

Induced subgraph H of PDG ( { p 1 , , p 12 } ) where, for better visibility, the edges between A 1 and A 3 are “hidden” behind the edges between A 1 and A 2 and the edges between A 2 and A 3 .

3.2 The divisibility graph of a commutative ring

Definition 3.6

Let R be a commutative ring and a, b be distinct elements of R.

  1. If a is a non-zero non-unit element of R, then we say a is a proper element of R.

  2. If a | b (in R) and b a (in R), then we write a | | b .

  3. The divisibility graph Div ( R ) of R is the undirected simple graph whose vertex set consists of the proper elements of R such that two vertices a b are adjacent if and only if a | | b or b | | a .

The following lemma can be verified by a straight-forward argument.

Lemma 3.7

Let R be a commutative ring and let V be the set of all proper elements of R and define on V such that for all a, b V , we have a b if and only if a = b or a | | b .

Then ( V , ) is a partially ordered set and the divisibility graph Div ( R ) of R is a partial order graph.

By Lemma 3.7, it is clear that D i v G ( n , m ) P o G ( n , m ) holds. However, since a perfect divisor graph is an induced subgroup of a divisibility graph, it follows from Theorem 3.4 that equality holds. We conclude the following theorem.

Theorem 3.8

Let n, m 1 be positive integers (n, m need not be distinct). Then for the Ramsey number D i v G with respect to the class D i v G of divisibility graphs the following holds:

D i v G ( n , m ) = P o G ( n , m ) = ( n 1 ) ( m 1 ) + 1 .

Moreover, in view of Theorem 3.8, we have the following result.

Corollary 3.9

Let n, m 1 be positive integers (n, m need not be distinct), k = ( n 1 ) ( m 1 ) + 1 , R be a commutative ring and S be a subset of proper elements of R such that | S | k .

Then one of the following assertions holds:

  1. There are n elements a 1 , , a n S such that a 1 | | a 2 | | | | a n (in R).

  2. There are m pairwise distinct elements b 1 , , b m S such that for all 1 h f m either

    • b h b f or

    • b h | b f and b f | b h

hold.

3.3 Inclusion ideal graphs of rings

Definition 3.10

Let R be a ring.

  1. We call a left (right) ideal I of R non-trivial if I { 0 } and I R .

  2. The inclusion ideal graph In ( R ) of R is the (simple, undirected) graph whose vertex set is the set of non-trivial left ideals of R and two distinct left ideals I, J are adjacent if and only if I J or J I (cf. Akbari et al. [10]).

  3. By I n G , we denote the class of all inclusion ideal graphs.

Remark 3.11

The set V of all non-trivial left ideals of a ring R together with the partial order induced by inclusion is a partially ordered set. Hence, the inclusion graph In ( R ) of a ring R is a partial order graph.

By Remark 3.11, it is clear that I n G ( n , m ) P o G ( n , m ) . The reverse inequality can be seen from the following argument. Let G be the graph constructed in the proof of Theorem 3.4, that is, G = PDG ( S ) with S = { p 1 , , p w } is the set of w = ( n 1 ) ( m 1 ) distinct positive primes of Z . Recall that G contains a subgraph with ( n 1 ) ( m 1 ) vertices whose clique number is at most n 1 and in which not more than m 1 vertices are independent. The graph G is graph-isomorphic to a subgraph of the inclusion ideal graph of Z , namely, the subgraph induced by the principal ideals generated by the elements in the vertex set of G. Since the inclusion ideal graph of Z is contained in I n G , it follows that ( n 1 ) ( m 1 ) < I n G ( n , m ) . Hence, by Theorem 2.2 we conclude the following theorem.

Figure 5 
                  Subgraph H of 
                        
                           
                           
                              
                                 
                                    ConeG
                                 
                                 
                                    3
                                 
                              
                              (
                              Z
                              )
                           
                           {\text{ConeG}}_{3}({\mathbb{Z}})
                        
                      induced by the vertices 
                        
                           
                           
                              V
                              =
                              {
                              1
                              ,
                              2
                              ,
                              3
                              ,
                              …
                              ,
                              12
                              }
                           
                           V=\{1,2,3,\ldots ,12\}
                        
                     .

Figure 5

Subgraph H of ConeG 3 ( Z ) induced by the vertices V = { 1 , 2 , 3 , , 12 } .

Theorem 3.12

Let n, m 1 be positive integers (n, m need not be distinct). Then for the Ramsey number I n G with respect to the class I n G of inclusion ideal graphs the following holds:

I n G ( n , m ) = P o G ( n , m ) = ( n 1 ) ( m 1 ) + 1 .

In view of Theorem 3.12, we have the following result.

Corollary 3.13

Let R be a ring, n, m 1 be positive integers (n, m need not be distinct) and S { I | I i s a n o n - t r i v i a l l e f t i d e a l o f R } such that | S | ( n 1 ) ( m 1 ) + 1 .

Then one of the following assertions hold:

  1. There are n pairwise distinct elements (non-trivial left ideals) I 1 , , I n S with I 1 I 2 I n .

  2. There are m elements (non-trivial left ideals) J 1 , , J m S such that J a J b for every 1 a b m .

3.4 Matrix graphs over commutative rings

Definition 3.14

Let R be a commutative ring which is not a field and j 2 an integer.

  1. We denote by R j × j the ring of all j × j matrices with entries in R.

  2. Let V = { A R j × j | det ( A ) a proper element of R } be the set of all j × j matrices whose determinant is a proper element of R, cf. Definition 3.6. We define the matrix graph Mat G ( R ) of R to be the undirected simple graph with V as its vertex set and two distinct vertices A, B V are adjacent if and only if det ( A ) | | det ( B ) or det ( A ) | | det ( B ) .

  3. By a t G we denote the class of all matrix graphs.

Lemma 3.15

Let R be a commutative ring which is not a field, j 2 an integer and

V = { A R j × j | det ( A ) i s a p r o p e r e l e m e n t o f R } .

Define on V such that for all A, B V , we have A B if and only if A = B or det ( A ) | | det ( B ) .

Then ( V , ) is a partially ordered set and the graph MatG ( R ) is a partial order graph.

By Theorem 2.2, it is clear that a t G ( n , m ) P o G ( n , m ) . We prove next that equality holds.

Theorem 3.16

Let n, m 1 be positive integers (n, m need not be distinct). Then for the Ramsey number a t G with respect to the class a t G of matrix graphs the following holds:

a t G ( n , m ) = P o G ( n , m ) = ( n 1 ) ( m 1 ) + 1 .

Proof

Let R = Z and j 2 and set w = ( n 1 ) ( m 1 ) 1 . Furthermore, let p 1 , p 2 , , p w be distinct positive prime numbers of Z and choose X i R j × j with det ( X i ) = p i for 1 i w .

We construct a matrix graph MatG ( R ) which has a complete ( n 1 ) -partite subgraph H in which each partition has m 1 vertices. The construction is analogous to the one in the proof of Theorem 3.4.

For each 1 i n 1 , let k i = ( i 1 ) ( m 1 ) , q i = X 1 X 2 X k i (hence q 1 = I j the identity matrix j × j ) and

A i = { q i X k i + 1 , , q i X k i + ( m 1 ) } .

Note that A 1 = { X 1 , , X m 1 } . Since det ( q i X k i + j ) = p 1 p k i p k i + j , it follows that the elements of A i are pairwise distinct and | A i | = m 1 for 1 i n 1 .

Let S = A 1 A 2 A n 1 and set G = MatG ( Z ) . Then for each i, the vertices in A i are independent. However, there are edges between all vertices of two distinct sets A i and A j with i j . Therefore, G is a complete ( n 1) -partite graph in which each partition has m 1 vertices that are independent. Thus, at most m 1 vertices of G are independent. It is easily verified that the clique number of G is n 1 . It follows that a t G ( n , m ) > w and together with Theorem 2.2 we conclude a t G ( n , m ) = P o G ( n , m ) = w + 1 = ( n 1 ) ( m 1 ) + 1 .□

Corollary 3.17

Let R be a commutative ring, j 2 , n, m 1 be positive integers (n, m need not be distinct) and S { X D | det ( X ) b e a p r o p e r e l e m e n t o f R } such that | S | ( n 1 ) ( m 1 ) + 1 .

Then one of the following assertions hold:

  1. There are n matrices X 1 , , X n S such that det ( X 1 ) | | det ( X 2 ) | | | | det ( X n ) (in R).

  2. There are m pairwise distinct matrices Y 1 , , Y m S such that for all 1 h f m .

    • det ( Y h ) det ( Y f ) or

    • det ( Y h ) | det ( Y f ) and det ( Y f ) | det ( Y h )

hold.

3.5 Idempotent graphs of commutative rings

Definition 3.18

Let R be a commutative ring.

  1. We call a R idempotent if a 2 = a .

  2. We define the idempotent graph Idm ( R ) of R to be the undirected simple graph with the set of idempotents of R as its vertex set and two distinct vertices a, b are adjacent if and only if a | b or b | a .

  3. By I d e m G we denote the class of all idempotent graphs.

First, we show that the divisibility relation is a partial order on the set of idempotent elements of R.

Lemma 3.19

Let R be a commutative ring and let V be the set of all idempotent elements of R. We define on V such that for all a, b V , we have a b if and only if a | b .

Then ( V , ) is a partially ordered set and the graph Idm ( R ) is a partial order graph.

Proof

Clearly, is reflexive and transitive. Suppose that a | b and b | a (in R), that is, a = b x and b = a y for some x, y R . Then, since a and b are idempotent, we can conclude that

a b a = ( 1 b ) a = ( 1 b ) b x = b x b 2 x = b x b x = 0 and b a b = ( 1 a ) b = ( 1 a ) a y = a y a 2 y = a y a y = 0

and hence a = b a = a b = b , which implies that is anti-symmetric.□

By Lemma 3.19, it is clear that I d e m G ( n , m ) P o G ( n , m ) . Next, we show that I d e m G ( n , m ) = P o G ( n , m ) . We start with the following lemma.

Lemma 3.20

Let R be a commutative ring and E be a set of w 3 distinct non-trivial idempotents of R such that eR is a maximal ideal of R for every e E . Let x = f 1 f 2 f k and y = b 1 b 2 b j such that f 1 , , f k , b 1 , , b j E and 2 k , j < w .

Then

  1. x 0 .

  2. x = y if and only if { f 1 , , f k } = { b 1 , , b j } .

Proof

  1. Since e 1 , , e w are distinct non-trivial idempotents of R and each e i R is a maximal ideal of R, 1 i w , by Lemma 3.19 we conclude that e 1 R , , e w R are distinct maximal ideals of R. Since k < w , there exists a maximal ideal dR for some d E such that x = f 1 f 2 f k d R (note that each f i R is a maximal ideal of R). Thus, x 0 .

  2. We may assume that f 1 b i for every 1 i j . Hence, x f 1 R but y f 1 R and thus x y . Since all f i and b i are idempotent elements, multiplicities have no impact, which makes the other implication obvious.□

Theorem 3.21

Let n, m 1 be positive integers (n, m need not be distinct). Then for the Ramsey number I d e m G ( n , m ) with respect to the class of idempotent graphs the following holds:

I d e m G ( n , m ) = P o G ( n , m ) = ( n 1 ) ( m 1 ) + 1 .

Proof

We set w = ( n 1 ) ( m 1 ) 1 and show that I d e m G contains an ( n 1 ) -partite graph in which each partition consists of m 1 independent vertices. For this purpose, set R = i = 1 w Z 2 . It is clear that R has exactly w distinct maximal ideals, say M 1 , , M w , and each M i = p i R , 1 i w for idempotent p i of R. We set E = { p 1 , p 2 , , p w } . Note that | E | = w since p 1 , p 2 , , p w are pairwise distinct.

For each 1 i n 1 , let n i = ( i 1 ) ( m 1 ) , a i = p 1 p 2 p n i (hence a 1 = 1 ) and A i = { a i p n i + 1 , , a i p n i + ( m 1 ) } . Note that A 1 = { p 1 , , p m 1 } .

By construction of each A i and in light of Lemma 3.20, for each 1 i n 1 , we have | A i | = m 1 and the vertices of A i are independent. Let H be the subgraph of I d m ( R ) , which is induced by A 1 A 2 A n 1 .

By construction of H and Lemma 3.20, we conclude that H is a complete ( n 1 ) -partite graph in which each partition has m 1 vertices that are independent. Thus, H has exactly m 1 vertices that are independent. It is easily verified that the clique number of H is n 1 . Thus, I d e m G ( n , m ) > w . Hence by Theorem 2.2, we have I d e m G ( n , m ) = P o G ( n , m ) = w + 1 = ( n 1 ) ( m 1 ) + 1 .□

Remark 3.22

Observe that the ring R = i =1 w Z 2 in the proof of Theorem 3.21 is a finite boolean ring. Let B o o l G denote the subclass of I d e m G consisting of all idempotent graphs of boolean rings.

In view of the proof of Theorem 3.21, we conclude that B o o l G ( n , m ) = I d e m G ( n , m ) . Thus, we state this result without a proof.

Theorem 3.23

Let n , m 1 be positive integers ( n , m need not be distinct).

Then B o o l G ( n , m ) = I d e m G ( n , m ) = P o G ( n , m ) = ( n 1 ) ( m 1 ) + 1 .

In view of Theorem 3.21, we have the following result.

Corollary 3.24

Let n, m 1 be positive integers (n, m need not be distinct), k = ( n 1 ) ( m 1 ) + 1 and A be a subset of idempotent elements of R such that | A | k .

Then one of the following assertions hold:

  1. There are n pairwise distinct elements (distinct idempotents) a 1 , , a n A such that a 1 | a 2 | | a n (in R).

  2. There are m pairwise distinct elements (distinct idempotents) b 1 , , b m A such that b h b f (in R) for all 1 h f m .

4 An example class C of partial order graphs with C ( n , m ) C ( m , n )

In this section, we present a subclass C of P D G with respect to which the Ramsey numbers C are non-symmetric in m and n. We recall the following definition [11].

Definition 4.1

A subset S of a ring R is called a positive semi-cone of R if S satisfies the following conditions:

  1. S ( S ) = { 0 } .

  2. S + S S .

  3. S S S .

If S satisfies the aforementioned conditions and S ( S ) = R , then S is called a positive cone of R [12].

For a positive semi-cone S of R, define S on R such that for all a, b R , we have a S b if and only if b a S . Then ( R , S ) is a partially ordered set. We define the S-positive semi-cone graph ConeG S ( R ) of R to be the simple, undirected graph with vertex set R such that two vertices a, b are connected by an edge if and only if b a S or a b S . Then Cone G S ( R ) is a partial ordered graph.

Definition 4.2

For k 2 , let P k = { 0 , k , 2 k , 3 k , } = k 0 .

  1. For a, b Z we define a k b if and only if b a P k .

  2. We define the k-positive semi-cone graph ConeG k ( Z ) of Z to be the simple, undirected graph with vertex set Z such that two vertices a, b R are connected by an edge if and only if | a b | P k .

  3. For every positive integer k 2 , let k - C o n e G ( n , m ) be the minimal number of vertices r such that every induced subgraph of the partial order graph ConeG k ( Z ) consisting of r vertices contains either a complete n-clique K n or an independent set consisting of m vertices.

Remark 4.3

  1. For every k 2 , P k is a positive semi-cone subset of Z that is not a positive cone of Z . The relation a k b if and only if b a P k is a partial order on Z and ConeG k ( Z ) is a partial order graph.

  2. For every k 2 , two vertices a, b of ConeG k ( Z ) are connected by an edge if and only if a b ( mod k ) .

For each k 2 , the following theorem shows that k - C o n e G ( n , m ) is not always symmetric in m and n.

Theorem 4.4

Let k 2 , n, m 1 be positive integers (n, m need not be distinct). Then

  1. If 1 m k + 1 , then

    k - C o n e G ( n , m ) = ( n 1 ) ( m 1 ) + 1 .

    In particular, if 1 n , m k + 1 , then k - C o n e G ( n , m ) = k - C o n e G ( m , n ) = ( n 1 ) ( m 1 ) + 1 is symmetric in n and m.

  2. If m > k + 1 , then

k - C o n e G ( n , m ) = k - C o n e G ( n , k + 1 ) = ( n 1 ) k + 1

only depends on the first argument n. In particular, assume that n m . If n > k + 1 or m > k + 1 , then k - C o n e G ( n , m ) k - C o n e G ( m , n ) .

Proof

(1) For n = 1 or m = 1 , the assertion immediately follows, so we assume n 2 and 2 m k + 1 . For each 1 i m 1 , let

A i = { k + i , 2 k + i , , ( n 1 ) k + i } .

By construction, each A i contains n 1 distinct elements a with a i P k . Therefore, for a b A i , either b a P k or a b P k and hence each A i induces a complete subgraph of ConeG k ( Z ) with exactly n 1 vertices. Moreover, since m 1 k , for a A i and b A j with 1 i j m 1 , then a b ( mod k ) and therefore a and b are not connected by an edge.

Let H be the subgraph of ConeG k ( Z ) which is induced by the vertex set A 1 A m 1 . Then H is disjoint union of m 1 ( n 1) -cliques and hence does neither contain an n-clique nor an independent set of cardinality m, which implies that k - C o n e G ( n , m ) > ( n 1 ) ( m 1 ) . It now follows from Theorem 2.2 that k - C o n e G ( n , m ) = ( n 1 ) ( m 1 ) + 1 .

The symmetry assertion follows immediately from this if, moreover, 1 n k + 1 holds.

(2) Recall that two vertices a, b of ConeG k ( Z ) are connected by an edge if and only if a b ( mod k ) . Therefore, a maximal independent subset has cardinality k (the number of residue class modulo k). Thus if m k + 1 , then ConeG k ( Z ) cannot contain an independent set with m distinct vertices. Therefore, for all m k + 1 , the equality

k - C o n e G ( n , m ) = k - C o n e G ( n , k + 1 )

holds and the assertion now follows from (1).□

In view of Theorem 4.4, we have the following result.

Corollary 4.5

Let k 2 and n, m 1 be positive integers (n, m need not be distinct) and A be a subset of Z . Then

  1. If 2 m k + 1 and | A | > ( n 1 ) ( m 1 ) , then there are at least n pairwise distinct elements a 1 , , a n A such that a 1 a n ( mod k ) or there at least m elements b 1 , , b m A such that b i b j ( mod k ) for all 1 i j m .

  2. If m > k + 1 and | A | > ( n 1 ) k , then there are at least n pairwise distinct elements of A, say a 1 , , a n such that a 1 a n ( mod k ) .

Example 4.6

The induced subgraph H of ConeG 3 ( Z ) with vertex set V = { 1 , 2 , 3 , , 12 } consists of three 4-cliques. Since | V | = 12 , H satisfies 3 -Cone G ( 12 , 2 ) , 3 -Cone G ( 4 , 4 ) , 3 -Cone G ( 6 , 3 ) , 3 -Cone G ( 5 , 3 ) and 3 -Cone G ( 4 , 10 ) , see Figure 5.

Acknowledgments

Ayman Badawi is supported by the American University of Sharjah Research Fund (FRG2019): AS1602. Roswitha Rissner is supported by the Austrian Science Fund (FWF): P 28466. The authors would like to thank the referees for a careful reading of the paper.

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Received: 2020-06-26
Revised: 2020-10-15
Accepted: 2020-10-30
Published Online: 2020-12-31

© 2020 Ayman Badawi and Roswitha Rissner, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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