On two problems in Ramsey-Tur\'an theory

TL;DR

This paper investigates two problems in Ramsey-Turán theory, determining asymptotic maximums for edge-colorings and triangle counts in graphs with sub-linear independence number, revealing surprising differences from classical extremal constructions.

Contribution

It provides the first asymptotic results for these problems in graphs with sub-linear independence number, challenging previous assumptions about extremal structures.

Findings

The extremal construction does not maximize colorings with no monochromatic $K_k$ in sub-linear independence graphs.

Maximum number of triangles in $K_k$-free graphs with sub-linear independence number is asymptotically determined.

Extremal graphs resemble those maximizing edges in classical Ramsey-Turán problems.

Abstract

Alon, Balogh, Keevash and Sudakov proved that the $(k-1)$-partite Tur\'an graph maximizes the number of distinct $r$-edge-colorings with no monochromatic $K_k$ for all fixed $k$ and $r=2,3$, among all $n$-vertex graphs. In this paper, we determine this function asymptotically for $r=2$ among $n$-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with sub-linear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an $n$-vertex $K_k$-free graph $G$ with $\alpha(G)=o(n)$. The extremal graphs have similar structure to the extremal graphs for the classical Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.