Size multipartite Ramsey numbers for small paths versus books

Abstract

Given j ≥ 2, for graphs G and H, the size Ramsey multipartite number m j (G, H) is defined as the smallest natural number t such that any blue red coloring of the edges of the graph K j×t , necessarily containes a red G or a blue H as subgraphs. Let the book with n pages is defined as the graph K 1 + K 1,n and denoted by B n . In this paper, we obtain the exact values of the size Ramsey numbers m j (P 3 , H) for j ≥ 3 where H is a book B n . We also derive some upper and lower bounds for the size Ramsey numbers m j (P 4 , H) where H is a book B n .

Size Multipartite Ramsey Numbers for Small Paths vs. K2,n

“…Moreover, diagonal classical Ramsey number r(n,n) is defined as r(K n ,K n ). The exact determination of these diagonal classical Ramsey numbers have been studied for a few decades (see Radziszowski, 2017 for a survey) but sadly not much progress has been done even in the case of r (5,5). In the last decade, using this idea of the original classical Ramsey numbers and of the size Ramsey numbers, the notion of size multipartite Ramsey numbers were introduced by Burger and Vuuren (i.e., Burger et al, 2004) by exploring the two colourings of multipartite graph K j×s instead of the complete graph.…”