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2011
Springer

# Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

9 years 5 months ago
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
Tao Jiang, Michael Salerno
 Added 14 May 2011 Updated 14 May 2011 Type Journal Year 2011 Where GC Authors Tao Jiang, Michael Salerno