On the structural growth of bipartite Ramsey numbers

Abstract: Bipartite Ramsey numbers is the smallest size of a complete bipartite graph $K_{N,N}$ such that every edge-coloring with a given number of colors inevitably yields a monochromatic copy of a prescribed bipartite graph. While exact values have been determined for certain specific graphs, the general asymptotic behavior of these numbers in terms of structural graph parameters remains poorly understood. In this paper, we investigate structure-dependent growth phenomena in bipartite Ramsey theory. We first establish a general lower bound for the $q$-color bipartite Ramsey number $\operatorname{br}(K_{s,t};q)$. The proof employs a probabilistic construction together with an optimization over independent set densities, adapting the approach of Nikiforov and Sawin to the bipartite context. Next, for a fixed bipartite graph $G$ with $p$ vertices and $q$ edges, we prove a lower bound of the form $\operatorname{br}(G,K_{n,n}) &gt; C \bigl(\frac{n}{\log n}\bigr)<sup>{(q-1)/(p-2)}$. As a corollary, we show that sufficiently dense bipartite graphs fail to be bipartite Ramsey size linear. Turning to even cycles and complete bipartite graphs, we obtain an upper bound on the multicolor bipartite Ramsey number $$\operatorname{br}<em>k(C</em>{2t};K_{n,n}) \le c_{t,k}\, n^2/\log² n$$, which follows from classical estimates for Zarankiewicz numbers together with a double-counting argument. Building on this result, we further derive a refined linear upper bound of the form $$\operatorname{br}(C_{2t},G) \le \frac{m}{2} + \frac{29t\sqrt{m}}{2}$$, valid for any connected bipartite graph $G$ with $m$ edges and no isolated vertices.