Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

The Ramsey number r(H, Kn) is the smallest positive integer N such that every graph of order N contains either a copy of H or an independent set of size n. The Tur´an number ex(m, H) is the maximum number of edges in a graph of order m not containing a copy of H. We prove the following two results: (1) Let H be a graph obtained from a tree F of order t by adding a new vertex w and joining w to each vertex of F by a path of length k such that any two of these paths share only w. Then r(H, Kn) ≤ ck,t n1+1/k ln1/k n , where ck,t is a constant depending only on k and t. This generalizes some results in [11], [13], and [16]. (2) Let H be a bipartite graph with ex(m, H) = O(mγ), where 1 < γ < 2. Then r(H, Kn) ≤ cH n ln n 1/(2−γ) , where cH is a constant depending only on H. This generalizes a result in [4]. Key words. Ramsey number, independence number

• Type: Journal
• Year: 2011
• Where: GC
• Authors: Tao Jiang, Michael Salerno
• Keywords: Applied Computing, GC 2011, Independent Set, Ramsey Number, Smallest Positive Integer