Polynomially superlinear growth of set-coloring Ramsey numbers

Abstract: The set-coloring Ramsey number $R(k;r,s)$ is the least $N$ such that every assignment of an $s$-element subset of $[r]$ to each edge of $K_N$ yields a copy of $K_k$ whose edges share a common color. For every fixed prime power $q$, we construct infinitely many positive integer triples $(r,j,s)$ with $j\sim(q-1)<sup>{-2/3}r<sup>{1/3}$ and $s=(1-1/q)(r-j)$ such that $R(q+1;r,s)=Θ_q(r<sup>{4/3})$. For $q=3$, this answers in the affirmative a question of Conlon, Fox, Pham and Zhao, showing that polynomially superlinear growth for $R(4;r,2(r-j)/3)$ already occurs at the scale (j=Θ(r^{1/3})). Moreover, along the same sequence, the maximum size of a $q$-ary code of length $r$ and minimum Hamming distance at least $s$ is $(1+o(1))(q-1)<sup>{4/3}r<sup>{4/3}$.