A strict upper bound for size multipartite Ramsey numbers of paths versus stars - Chula Jayawardene; Lilanthi Samarasekara (Indonesian Combinatorial Society (InaCombS), 2017)
Let P_n represent the path of size n. Let K_1,m-1 represent a star of size m and be denoted by S_m. Given a two coloring of the edges of a complete graph K_j × s we say that K_j × s→ (P_n,S_m+1) if there is a copy of P_n in the first color or a copy of S_m+1 in the second color. The size Ramsey multipartite number m_j(P_n, S_m+1) is the smallest natural number s such that K_j × s→ (P_n,S_m+1). Given j,n,m if s=n+m-1-kj-1 ceil, in this paper, we show that the size Ramsey numbers m_j(P_n,S_m+1) is bounded above by s for k=n-1j ceil. Given j3 and s, we will obtain an infinite class (n,m) that achieves this upper bound s. In the later part of the paper, will also investigate necessary and sufficient conditions needed for the upper bound to hold.