Unavoidable chromatic patterns in 2-colorings of the complete graph

TL;DR

This paper investigates unavoidable edge-coloring patterns in complete graphs, characterizing graphs that always appear with any 2-coloring and establishing bounds on the number of edges needed to guarantee such patterns.

Contribution

It provides a structural characterization of omnitonal graphs, showing they are bipartite, and derives bounds on the function ${ m ot}(n,G)$ related to Turán numbers.

Findings

Omnitonal graphs are bipartite.

Bounds on ${ m ot}(n,G)$ depend on graph parameters.

Certain graphs satisfy ${ m ot}(n,G) = ex(n,G)$.

Abstract

We consider unavoidable chromatic patterns in $2$-colorings of the edges of the complete graph. Several such problems are explored being a junction point between Ramsey theory, extremal graph theory (Tur\'an type problems), zero-sum Ramsey theory, and interpolation theorems in graph theory. A role-model of these problems is the following: Let $G$ be a graph with $e(G)$ edges. We say that $G$ is omnitonal if there exists a function ${\rm ot}(n,G)$ such that the following holds true for $n$ sufficiently large: For any $2$-coloring $f: E(K_n) \to \{red, blue \}$ such that there are more than ${\rm ot}(n,G)$ edges from each color, and for any pair of non-negative integers $r$ and $b$ with $r+b = e(G)$, there is a copy of $G$ in $K_n$ with exactly $r$ red edges and $b$ blue edges. We give a structural characterization of omnitonal graphs from which we deduce that omnitonal graphs are, in particular, bipartite graphs, and prove further that, for an omnitonal graph $G$, ${\rm ot}(n,G) = \mathcal{O}(n^{2 - \frac{1}{m}})$, where $m = m(G)$ depends only on $G$. We also present a class of graphs for which ${\rm ot}(n,G) = ex(n,G)$, the celebrated Tur\'an numbers. Many more results and problems of similar flavor are presented.