Growth rates of the bipartite Erdős–Gyárfás function

Given two graphs (Formula presented.) and a positive integer (Formula presented.), an (Formula presented.) -coloring of (Formula presented.) is an edge-coloring of (Formula presented.) such that every copy of (Formula presented.) in (Formula presented.) receives at least (Formula presented.) distinct colors. The bipartite Erdős–Gyárfás function (Formula presented.) is defined to be the minimum number of colors needed for (Formula presented.) to have a (Formula presented.) -coloring. For balanced complete bipartite graphs (Formula presented.), the function (Formula presented.) was studied systematically in Axenovich et al. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs (Formula presented.) that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi.