Erdős–Szekeres Convex Polygon Conjecture

For $k\ge3$, let $f(k)$ denote the least $N$ such that every set of $N$ points in the plane with no three collinear contains $k$ points forming a convex polygon. Erdős and Szekeres proved in 1935 that $f(k)$ is finite, in a paper that helped launch both Ramsey theory and combinatorial geometry, and conjectured the exact value $f(k)=2^{k-2}+1$ [ErdosSzekeres1935].

The lower bound is settled: a construction of Erdős and Szekeres exhibits $2^{k-2}$ points in general position with no $k$ points in convex position, so $f(k)\ge2^{k-2}+1$, and this is known to be sharp for small values of $k$. The upper bound proved much harder. The original argument gave a bound of roughly $4^k$, and for eighty years improvements affected only lower-order factors. Suk then showed $f(k)\le2^{k+o(k)}$, determining the answer up to sub-exponential factors [Suk2017HappyEnding].

The conjecture thus stands confirmed in the exponent but open in its exact form: a resolution requires eliminating the $o(k)$ error term and pinning $f(k)$ down to exactly $2^{k-2}+1$ for every $k$.