STAR-CRITICAL RAMSEY NUMBERS AND REGULAR RAMSEY NUMBERS FOR STARS

Let G be a graph, H be a subgraph of G, and let G - H be the graph obtained from G by removing a copy of H. Let K-1,K- n be the star on n + 1 vertices. Let t >= 2 be an integer and H-1, ..., H-t and H be graphs, and let H -> (H-1, ..., H-t) denote that every t coloring of E(H) yields a monochromatic copy of H-i in color i for some i is an element of[t]. The Ramsey number r(H-1, ..., H-t) is the minimum integer N such that K-N -> (H-1, ..., H-t). The star-critical Ramsey number r(*)(H-1, ..., H-t) is the minimum integer k such that K-N - K-1,K- N-1-k -> (H-1, ..., H-t) where N = r(H-1, ..., H-t). Let rr(H-1, ..., H-t) be the regular Ramsey number for H-1, ..., H-t, which is the minimum integer r such that if G is an r-regular graph on r(H-1, ..., H-t) vertices, then G -> (H-1, ..., H-t). Let m(1), ..., m(t) be integers larger than one, exactly k of which are even. In this paper, we prove that if k >= 2 is even, then r(*)(K-1,K- m1, ..., K-1,K- mt) = Sigma(t)(i=1) m(i) - t + 1/2 which disproves a conjecture of Budden and DeJonge in 2022. Furthermore, we prove that rr(K-1,K- m1, ..., K-1,K- mt) = {Sigma(t)(i=1) m(i) - t, k >= 2 is even, Sigma(t)(i=1) m(i) - t + 1, otherwise.