Unavoidable chromatic patterns in 2‐colorings of the complete graph.

Given a graph G on k edges, we consider the following two extremal problems: provided n is large enough, what is the minimum integer bal(n,G), if it exists, such that any 2‐coloring of the edges of a complete graph on n vertices having more than bal(n,G) edges in each color class, contains a balanced copy of G, that is, a copy of G with exactly ⌊k∕2⌋ edges in one of the colors? Graphs for which this is possible are called balanceable. The second problem deals with a similar question but we seek to guarantee copies of G in every tone, that is, having exactly r edges in, say, color red, for every 0≤r≤k. Graphs with this property are called omnitonal. We study these problems for different graph families, including paths, stars, and trees in general. When studying such extremal parameters, the question of its existence is obliged. In this line, two universal unavoidable patterns in 2‐edge‐colorings of the complete graph with sufficient representation in each of the colors emerge naturally and they are the key to characterizing balanceable as well as omnitonal graphs. For the two universal unavoidable patterns, which were already known to exist via a Ramsey‐theoretic approach, we present here a Turán‐type counterpart. [ABSTRACT FROM AUTHOR]