On anti-Ramsey numbers for complete bipartite graphs and the Turan function

TL;DR

This paper investigates the relationship between anti-Ramsey numbers and Turán functions for complete bipartite graphs, establishing bounds that connect these two graph invariants and extending classical extremal graph theory results.

Contribution

It proves that the difference between the anti-Ramsey number and the Turán function for complete bipartite graphs is bounded by a linear function of n, linking these concepts and deriving new bounds.

Findings

The difference AR(K_n,K_{s,t}) - ex(K_n,K_{s,t}) is bounded by a constant times n.

AR(K_n,K_{s,t}) is at most proportional to n^{2 - 1/s}.

The bounds depend only on parameters s and t, not on n.

Abstract

Given two graphs $G$ and $H$ with $H\subseteq G$ we consider the anti-Ramsey function $AR(G,H)$ which is the maximum number of colors in any edge-coloring of $G$ so that every copy of $H$ receives the same color on at least one pair of edges. The classical Tur\'an function for a graph $G$ and family of graphs $\mathcal{F}$, written $ex(G,\mathcal{F})$, is defined as the maximum number of edges of a subgraph of $G$ not containing any member of $\mathcal{F}$. We show that there exists a constant $c>0$ so that $AR(K_n,K_{s,t})-ex(K_n,K_{s,t})<cn$ and $c$ depends only on $s$ and $t$, which implies $AR(K_n,K_{s,t})\leq cn^{2-\frac{1}{s}}$, for $s\leq t$ by a result of K\H ovari, S\'os, and Tur\'an.