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 $n$.