A strict upper bound for size multipartite Ramsey numbers of paths versus stars

Abstract

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 − k j − 1 , 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 − 1 j . Given j ≥ 3 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.

On size multipartite Ramsey numbers for stars

“…Furthermore, we gave the size multipartite Ramsey numbers for stars versus cycles [5] and the size tripartite Ramsey numbers for a disjoint union of m copies of a star K 1,n versus P 3 [6]. In 2017, Jayawardene et al [4] and Effendi et al [2] determined the size multipartite Ramsey numbers for stars versus paths. Then, we also gave the size multipartite Ramsey numbers for stars versus paths and cycles [7], that complete the previous results given by Syafrizal and Surahmat.…”