Unavoidable patterns in $2$-colorings of the complete bipartite graph

We determine the colored patterns that appear in any $2$-edge coloring of Kn,n, with n large enough and with sufficient edges in each color. We prove the existence of a positive integer z₂ such that any $2$-edge coloring of Kn,n with at least z₂ edges in each color contains at least one of these patterns. We give a general upper bound for z₂ and prove its tightness for some cases. We define the concepts of bipartite r-tonality and bipartite omnitonality using the complete bipartite graph as a base graph. We provide a characterization for bipartite r-tonal graphs and prove that every tree is bipartite omnitonal. Finally, we define the bipartite balancing number and provide the exact bipartite balancing number for paths and stars.