  Given positive integers $p$ and $q$, a $(p,q)$-coloring of the complete graph
$K_n$ is an edge-coloring in which every $p$-clique receives at least $q$
colors. Erd\H{o}s and Shelah posed the question of determining $f(n,p,q)$, the
minimum number of colors needed for a $(p,q)$-coloring of $K_n$. In this paper,
we expand on the color energy technique introduced by Pohoata and Sheffer to
prove new lower bounds on this function, making explicit the connection between
bounds on extremal numbers and $f(n,p,q)$. Using results on the extremal
numbers of subdivided complete graphs, theta graphs, and subdivided complete
bipartite graphs, we generalize results of Fish, Pohoata, and Sheffer, giving
the first nontrivial lower bounds on $f(n,p,q)$ for some pairs $(p,q)$ and
improving previous lower bounds for other pairs.
