Lower bounds on the Erdős–Gyárfás problem via color energy graphs

Given positive integers (Formula presented.) and (Formula presented.), a (Formula presented.) -coloring of the complete graph (Formula presented.) is an edge-coloring in which every (Formula presented.) -clique receives at least (Formula presented.) colors. Erdős and Shelah posed the question of determining (Formula presented.), the minimum number of colors needed for a (Formula presented.) -coloring of (Formula presented.). 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 (Formula presented.). 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 (Formula presented.) for some pairs (Formula presented.) and improving previous lower bounds for other pairs.