Multicolor bipartite Ramsey numbers for quadrilaterals and stars

For bipartite graphs H1,…,Hμ, μ≥2, the μ-color bipartite Ramsey number, denoted by Rb(H1,…,Hμ), is the least positive integer N such that if we arbitrarily color the edges of a complete bipartite graph KN,N with μ colors, then it contains a monochromatic copy of Hi in color i for some i, 1≤i≤μ. Let C4 and K1,n be a quadrilateral and a star on n+1 vertices, respectively. In this paper, we show that the (μ+1)-color bipartite Ramsey number Rb(C4,…,C4,K1,n)≤n+⌈12μ2(4n+μ2+2μ−7)+4⌉+μ2+μ2−1. Moreover, using algebraic methods, we construct Ramsey graphs or near Ramsey graphs and determine infinitely many values of Rb(C4,…,C4,K1,n), which reach the upper bound if μ=1,2 and are at most ⌊μ2⌋ less than the upper bound if μ≥3.