Polynomial-to-Exponential Transition in Hypergraph Ramsey Numbers

Let r_k(s, e; t) denote the smallest N such that any red/blue edge coloring of the complete k-uniform hypergraph on N vertices contains either e red edges among some s vertices, or a blue clique of size t. Erdős and Hajnal introduced the study of this Ramsey number in 1972 and conjectured that for fixed s > k, there is a well defined value h_k(s) such that r_k(s, h_k(s)-1; t) is polynomial in t, while r_k(s, h_k(s); t) is exponential in a power of t. Erdős later offered $500 for a proof. Conlon, Fox, and Sudakov proved the conjecture for k=3 and 3-adically special values of s, and Mubayi and Razborov proved it for k at least 4. We prove the conjecture for k=3 and all s, settling all remaining cases of the problem. Joint work with Ruben Ascoli and Hans Yu.