Ramsey-Turán type results for matchings in edge colored graphs

Abstract

Let G1,G2,…,Gc be arbitrary graphs. By G→(G1,G2,…,Gc), we mean if the edges of G are partitioned into c disjoint color classes giving c graphs H1,H2,…,Hc, then at least one Hi has a subgraph isomorphic to Gi. The Ramsey number R(G1,G2,…,Gc) is the smallest positive integer n such that Kn→(G1,G2,…,Gc). In this paper, we extend the result of Cockayne and Lorimer on the Ramsey number of stripes allowing host graphs with the corresponding Ore-type condition: If t≥1 and n1,n2,…,nc, n1=max⁡{n1,n2,…,nc}, are positive integers and G is a graph with at least max⁡{t,n1}+∑i=1c(ni−1)+1 vertices such that for each pair of non-adjacent vertices, the sum of the number of their non-neighbors is at most 2max⁡{t,n1}−1, then G→(n1K2,n2K2,…,ncK2). Consequently, we prove that if G is a graph with chromatic number at least max⁡{t,n1}+∑i=1c(ni−1)+1, then G→(St,n1K2,n2K2,…,ncK2), where St is the star graph with t edges. In addition, for given graph G, the Ramsey closure of G, denoted by C(G), is defined and we prove that if G is a graph with at least max⁡{t,n1}+∑i=1c(ni−1)+1 vertices, then G→(n1K2,n2K2,…,ncK2) if and only if C(G)→(n1K2,n2K2,…,ncK2). Thus, a condition that forces C(G)→(n1K2,n2K2,…,ncK2) also forces G→(n1K2,n2K2,…,ncK2). As a result, a condition is provided on the degree sequence of a graph G such that under this condition G→(n1K2,n2K2,…,ncK2).Further, this article provides a sharp bound for the maximum number of edges in a simple graph G such that G↛(n1K2,n2K2,…,ncK2). We also establish the uniqueness of such extremal graphs. This result provides a far-reaching generalization of an important classical result of Erdős and Gallai and also giving a new proof for the graph case of the conjecture of Erdős dating back to 1965, known as the Erdős' matching conjecture.