Induced Graph Ramsey Theory

Abstract

We say that a graph \(F\) strongly arrows \((G,H)\) and write \(F \longmapsto (G,H)\) if for every edge-coloring of \(F\) with colors red and blue, a red \(G\) or a blue \(H\) occurs as an induced subgraph of \(F\). Induced Ramsey numbers are defined by \(r^*(G,H) = \min\{|V(F)| : F \longmapsto (G,H)\}\).
The value of \(r^*(G,H)\) is finite for all graphs, and good upper bounds on induced Ramsey numbers in general, and for particular families of graphs are known. Most of these results, however, use the probabilistic method, and therefore do not yield explicit constructions. This paper provides several constructions for upper bounds on \(r^*(G,H)\), including:\(r^*(C_n) = r^*(C_n,C_n) \leq c^{(logn)^2}\), \(r^*(T,K_n) \leq |T|n^{|T|log|T|}\), \(r^*(B,C_n) \leq |B|^{\lceil log n \rceil +4}\) ,where \(T\) is a tree, \(B\) is bipartite, \(K_n\) is the complete graph on \(n\) vertices, and \(C_n\) is a cycle on \(n\) vertices. We also have some new upper bounds for small graphs: \(r^*(K_3 + e) \leq 21\), and \(r^*(K_4 – e) \leq 46\).