Counting edge-colorings of a complete graph avoiding a rainbow $K_4$

Source abstract

For $k, r, n$ natural numbers let $ρ_{r,k}(K_n)$ be the number of $r$-edge-colorings of $K_n$ that do not contain a rainbow copy of a $K_k$, that is, a copy of $K_k$ in which all edges receive different colors. When $k=3$, the quantity $ρ_{r,3}(K_n)$ represents the number of Gallai Colorings. It was proved by Balogh and Li and independently by Bastos, Benevides and Han, that most of the Gallai colorings are 2-colorings, for $n$ large. A natural analogue conjecture would be that when $k=4$, $r\ge 5$ and $n$ large, most rainbow-$K_4$-free $r$-edge-colorings are $5$-colorings. We show that this is not true in general and identify an exact threshold for $r$ where this ceases to be true. For the range where the conjecture is false, we determine the exponential growth of $ρ_{r,4}(K_n)$ for every fixed $r$. More precisely, for $6\le r\le24$, we prove that $ρ_{r,4}(K_n)=(\binom{r}{5}+o(1))5^{\binom{n}{2}}$; and for each $r\ge25$, the proportion using at most five colors tends to zero, and $ρ_{r,4}(K_n)=r^{(n^2/4)+o(n^2)}$. A bipartite construction, with all edges within the two parts assigned one common color, achieves the latter exponential growth rate. The lower bounds can be easily generalized for every $k$. Those results are related to other recent results about counting colorings that avoid rainbow cliques or given rainbow patterns in general. Our proof combines hypergraph containers with the graph removal lemma, structural estimates for color palettes and a refined count of colorings close to a fixed five-color palette.