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

Abstract For a partially ordered set ( A , ≤ ) (A,\le ) , let G A {G}_{A} be the simple, undirected graph with vertex set A such that two vertices a ≠ b ∈ A a\ne b\in A are adjacent if either a ≤ b a\le b or b ≤ a b\le a . We call G A {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 G={G}_{A} . For a class C {\mathcal{C}} of simple, undirected graphs and n, m ≥ 1 m\ge 1 , we define the Ramsey number ℛ C ( n , m ) { {\mathcal R} }_{{\mathcal{C}}}(n,m) with respect to C {\mathcal{C}} to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in C {\mathcal{C}} consisting of r vertices contains either a complete n-clique K n {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.


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 of simple, undirected graphs and n, ≥ m 1, we define the Ramsey number ( ) n m , with respect to the class to be the minimal number r of vertices such that every induced subgraph of an arbitrary graph in 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 ( ( and are vertices of . 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 ( ) n m , o for the class o of partial order graphs equals ( − )( − ) + n m 1 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 of perfect divisor graphs, iv of divisibility graphs, In of inclusion ideal graphs, at of matrix graphs and Idem of idempotent graphs of rings, the respective Ramsey numbers equal to o , 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, and n will denote the integers and integer modulo n, respectively. Moreover, for a ring R we assume that ≠ 1 0 holds, 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 o we denote the class of all partial order graphs.
(2) For a class of simple, undirected graphs and n, ≥ m 1, we set ( ) n m , to be the minimal number r of vertices such that every induced subgraph of an arbitrary graph in consisting of r vertices contains either a complete n-clique K n or an independent set consisting of m vertices. We call the Ramsey number with respect to the class .
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 m 1 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 ( ) a pos H 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 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 of graphs, denotes the Ramsey number with respect to , cf. Definition 2.1.

Perfect divisor graphs
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 m j J j for ∅ ≠ ⊆ { … } J n 1, , . We show that for every for some a, ∈ c R. Therefore, d and m i are coprime elements of R which in particular implies that ∤ m d i . It follows that if d 1 and d 2 are distinct perfect divisors of m and | d d is a partially ordered set.
Moreover, it follows that the elements in V correspond to the non-empty proper subset of { … } n 1, , . Therefore, their number amounts to ( ) and ( ) S PDG the perfect divisor graph of m with respect to S. Then the following assertions hold: , then for the girth of ( ) S PDG the following holds 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 m j J j and = ∏ ∈ b m k K k be two distinct vertices of ( ) , be two distinct vertices of ( ) S PDG . In light of the proof given in (1) . Hence, every vertex of ( ) S PDG is connected by an edge to either one of these two vertices.
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 (6) For = n 3, we can verify in Figure 1 that there is cycle of length 6 and no shorter cycle.
If Finally, for (7), it is easily verified that ( ) S PDG is planar if = n 3, cf. Figure 1. Moreover, Figure 2 shows a planar arrangement of the edges of ( ) If, however, ≥ n 5, then Figure 3 is a K 3,3 subgraph of ( ) S PDG , and hence ( ) S PDG is not a planar by Kuratowski's Theorem on planar graphs. □       The subgraph of ( ) 3 is a complete 3-partite graph in which each partition has four vertices that are independent (Figure 4). 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 .

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.
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. 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.    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 , namely, the subgraph induced by the principal ideals generated by the elements in the vertex set of G. Since the inclusion ideal graph of is contained in InG, it follows that ( − )( − ) < ( ) n m n m 1 1 ,

In
. Hence, by Theorem 2.2 we conclude the following theorem.

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

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.
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 ( ) G R Mat 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 . We prove next that equality holds.
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.
the identity matrix × j j) and 1. Then one of the following assertions hold: First, we show that the divisibility relation is a partial order on the set of idempotent elements of R.  such

Idempotent graphs of commutative rings
, ,     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 a A , , n 1 such that | |⋯| a a a n 1 2 (in R).
4 An example class of partial order graphs with   ( ) ≠ ( ) n m m n , , In this section, we present a subclass of with respect to which the Ramsey numbers are nonsymmetric in m and n. We recall the following definition [11]. 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 b S if and only if − ∈ b a S. Then ( ≤ ) R, S is a partially ordered set. We define the S-positive semi-cone graph ( ) R ConeG S 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 ( ) G R Cone S is a partial ordered graph.
if and only if − ∈ b a P k . (2) We define the k-positive semi-cone graph ( ) ConeG k of to be the simple, undirected graph with vertex set such that two vertices a, ∈ b R are connected by an edge if and only if | − | ∈ a b P 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 with exactly − n 1 vertices. Moreover, since − ≤ m k 1 , for ∈ a A i and ∈ b A j with ≤ ≠ ≤ − i j m 1 1 , then ≢ ( ) a b k mod and therefore a and b are not connected by an edge.
Let H be the subgraph of ( ) ConeG k which is induced by the vertex set ∪ ⋯ ∪ . The symmetry assertion follows immediately from this if, moreover, ≤ ≤ + n k 1 1holds.
(2) Recall that two vertices a, b of ( ) ConeG k are connected by an edge if and only if ≡ ( ) a b k mod . Therefore, a maximal independent subset has cardinality k (the number of residue class modulo k). Thus if ≥ + m k 1, then ( ) ConeG k cannot contain an independent set with m distinct vertices. Therefore, for all ≥ + m k 1, the equality  , see Figure 5.