Denae Ventura

Extremal graph theory studies how the amount of edges in a graph can guarantee the existence of a particular substructure, and there are variants that involve edge colorings as well. It is known that any $2$-coloring of the edges of a large enough complete graph with enough edges in each color class contains at least one of two patterns, either a colored $K_{2t}$ where one color class induces a $K_t$ or a colored $K_{2t}$ where one color class induces two disjoint $K_{t}$s. This result has given birth to many interesting problems involving balanceability (which seeks structures with equal proportions of color) and omnitonality (which seeks structures with all possible proportions of color). In this talk, we will discuss some results concerning balanceability and the unavoidable pattern found when we color the edges of a large enough complete bipartite graph.