  Given integers $2\le t \le k+1 \le n$, let $g_k(t,n)$ be the minimum $N$ such
that every red/blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ yields
either a $(k+1)$-set containing $t$ red $k$-subsets, or an $n$-set with all of
its $k$-subsets blue. Erd\H{o}s and Hajnal proved in 1972 that for fixed $2\le
t \le k$, there are positive constants $c_1$ and $c_2$ such that $$ 2^{c_1 n} <
g_k(t, n) < twr_{t-1} (n^{c_2}),$$ where $twr_{t-1}$ is a tower of 2's of
height $t-2$. They conjectured that the tower growth rate in the upper bound is
correct. Despite decades of work on closely related and special cases of this
problem by many researchers, there have been no improvements of the lower bound
for $2<t<k$. Here we settle the Erd\H{o}s-Hajnal conjecture in almost all cases
in a strong form, by determining the correct tower growth rate, and in half of
the cases we also determine the correct power of $n$ within the tower.
Specifically, we prove that if $2<t<k-1$ and $k - t$ is even, then $$g_k(t, n)
= twr_{t-1} (n^{k-t+1 + o(1)}).$$ Similar results are proved for $k - t$ odd.
