Rainbow Tur\'an Methods for Trees

TL;DR

This paper investigates upper bounds for rainbow Turán numbers of specific trees using reduction methods, introduces the concept of k-unique colorings, and provides initial results for this new variant.

Contribution

It develops a reduction method for bounding rainbow Turán numbers and introduces k-unique colorings, expanding the scope of Turán-type problems for trees.

Findings

Upper bounds for rainbow Turán numbers of double stars, caterpillars, and binary trees.

Introduction of k-unique colorings and preliminary results.

Insights into the structure of rainbow-free colorings for trees.

Abstract

The rainbow Tur\'an number, a natural extension of the well studied traditional Tur\'an number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstra\"ete. The rainbow Tur\'an number of a graph $H$, $ex^{*}(n,H)$, is the largest number of edges for an $n$ vertex graph $G$ which can be properly edge colored with no rainbow $H$ subgraph. We explore the reduction method for finding upper bounds on rainbow Tur\'an numbers, and use this to inform results for the rainbow Tur\'an numbers of double stars, caterpillars, and perfect binary trees. In addition, we define $k$-unique colorings and the related $k$-unique Tur\'an numbers. We provide preliminary results on this new variant on the classic problem.