Conlon, Fox, Lee and Sudakov's Erdős–Gyárfás function conjecture

Let $f_r(n,p,q)$ denote the least number of colours in an $r$-uniform colouring of the complete hypergraph on $n$ vertices with no induced copy of a $p$-vertex hypergraph using at most $q-1$ colours. For positive integers $p$ and $r$ satisfying $2\leq r<p$, Conlon, Fox, Lee and Sudakov's conjecture.

$$f_r\left(n,p,\binom{p-1}{r-1}\right)=n^{o(1)}.$$

This conjecture extends the Erdős–Gyárfás problem from graphs to general uniformity. The surrounding results establish subpolynomial bounds for the cases $r=2$ and $(r,p)=(3,4)$, while polynomial lower bounds are known when the third parameter is increased to $\binom{p-1}{r-1}+1$; the general assertion remains open.