Polynomial-to-exponential transition in 3-uniform Ramsey numbers

Let rₖ (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. Erd os and Hajnal introduced the study of this Ramsey number in 1972 and conjectured that for fixed s>k 3, there is a well defined value hₖ (s) such that rₖ (s, hₖ (s) -1; t) is polynomial in t, while rₖ (s, hₖ (s) ; t) is exponential in a power of t. Erd os 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 s > k 4. We prove the conjecture for k=3 and all s, settling all remaining cases of the problem. We do this by solving a novel Turán-type problem: what is the maximum number of edges in an n-vertex 3-uniform hypergraph in which all tight components are tripartite? We show that the balanced iterated blowup of an edge is an exact extremizer for this problem for all n.