On the lower- and upper bounds of three-colored anti-Ramsey numbers

Abstract

In this paper we study a relatively new Ramsey type graph parameter introduced by my Bachelor's thesis supervisor Nóra Almási, and her Master's thesis supervisor Maria Axenovich. The parameter $ξ(G)$ denotes the largest natural number $n$, such that there is a red-blue-green edge coloring of $K_n$, such that in the edge coloring all copies of $G$ represent all three colors. After a brief recap on Ramsey numbers, we precisely define $ξ(G)$, and take a look at its relations to the Ramsey numbers. In the third chapter we establish lower bounds on $ξ(G)$, in the cases $G=K_5$ and $G=K_6$, namely $ξ(K_5)\ge 19$ and $ξ(K_6)\ge 31$.