Phase transitions of the Erdős-Gyárfás function
Xinyu Hu, Qizhong Lin, Xin Lu, Guanghui Wang
TL;DR
This work advances the study of the Erdős-Gyárfás function $f_k(n,p,q)$ for hypergraphs by providing explicit constructions that realize subpolynomial upper bounds in new regimes. It uses Mubayi-type colorings together with the Erdős–Hajnal stepping-up technique to build higher-uniformity colorings, culminating in a general $(k+2,3)$-coloring for all $k\ge4$. Concretely, it proves $f_k(n,k+2,3)=e^{O(\sqrt{\log_{(k-1)} n})}=(\log_{(k-2)} n)^{o(1)}$ for all $k\ge4$, and establishes the base case $k=4$ with $f_4(n,6,3)=e^{O(\sqrt{\log\log\log n})}=(\log\log n)^{o(1)}$, thereby confirming the conjectured transition in a new infinite family. The results illuminate the phase-transition behavior of $f_k$ and demonstrate how iterative colorings across uniformities can yield tight subpolynomial bounds with explicit constructions.
Abstract
Given positive integers $p,q$. For any integer $k\ge2$, an edge coloring of the complete $k$-graph $K_n^{(k)}$ is said to be a $(p,q)$-coloring if every copy of $K_p^{(k)}$ receives at least $q$ colors. The Erdős-Gyárfás function $f_k(n,p,q)$ is the minimum number of colors that are needed for $K_n^{(k)}$ to have a $(p,q)$-coloring. Conlon, Fox, Lee and Sudakov (\emph{IMRN, 2015}) conjectured that for any positive integers $p, k$ and $i$ with $k\ge3$ and $1\le i<k$, $f_k(n,p,{{p-i}\choose{k-i}})=(\log_{(i-1)}n)^{o(1)}$, where $\log_{(i)}n$ is an iterated $i$-fold logarithm in $n$. It has been verified to be true for $k=3, p=4, i=1$ by Conlon et. al (\emph{IMRN, 2015}), for $k=3, p=5, i=2$ by Mubayi (\emph{JGT, 2016}), and for all $k\ge 4, p=k+1,i=1$ by B. Janzer and O. Janzer (\emph{JCTB, 2024}). In this paper, we give new constructions and show that this conjecture holds for infinitely many new cases, i.e., it holds for all $k\ge4$, $p=k+2$ and $i=k-1$.