  A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ such that every
$p$-clique receives at least $q$ colors. In 1975, Erd\H{o}s and Shelah
introduced the generalized Ramsey number $f(n,p,q)$ which is the minimum number
of colors needed in a $(p,q)$-coloring of $K_n$. In 1997, Erd\H{o}s and
Gy\'arf\'as showed that $f(n,p,q)$ is at most a constant times
$n^{\frac{p-2}{\binom{p}{2} - q + 1}}$. Very recently the first author, Dudek,
and English improved this bound by a factor of $\log n^{\frac{-1}{\binom{p}{2}
- q + 1}} $ for all $q \le \frac{p^2 - 26p + 55}{4}$, and they ask if this
improvement could hold for a wider range of $q$.
  We answer this in the affirmative for the entire non-integral regime, that
is, for all integers $p, q$ with $p-2$ not divisible by $\binom{p}{2} - q + 1$.
Furthermore, we provide a simultaneous three-way generalization as follows:
where $p$-clique is replaced by any fixed graph $F$ (with $|V(F)|-2$ not
divisible by $|E(F)| - q + 1$); to list coloring; and to $k$-uniform
hypergraphs. Our results are a new application of the Forbidden Submatching
Method of the second and fourth authors.
