On the path-complete bipartite Ramsey number (Q583236)

An upper bound for the path-complete bipartite graph Ramsey number \(r(P_ k,K_{n,m})\leq n+m+k-2\) is proved. An immediate corollary is the exact result \(r(P_ k,K_{n,m})=n+m+k-2\) for \(n\equiv m\equiv 1 mod(k- 1),\) since \((n+m-3)/(k-1)\) disjoint copies of a \(K_{k-1}\) does not contain a \(P_ k\) and the complementary graph does not contain a \(K_{n,m}\). Clever use of a minimal counterexample type of argument gives a relatively short proof of the main inequality.