On the Erdős–Tuza–Valtr conjecture

The Erdős–Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erdős, Tuza, and Valtr strengthened the conjecture by stating that any set of more than ${\sum }_{i=n-b}^{a-2}\left(\frac{n-2}{i}\right)$ points in a plane either contains the vertices of a convex $n$-gon, $a$ points lying on a concave downward curve, or $b$ points lying on a concave upward curve. They also showed that the generalization is actually equivalent to the Erdős–Szekeres conjecture. We prove the first new case of the Erdős–Tuza–Valtr conjecture since the original 1935 paper of Erdős and Szekeres. Namely, we show that any set of $\left(\frac{n-1}{2}\right)+2$ points in the plane with no three points on a line and no two points sharing the same $x$-coordinate either contains 4 points lying on a concave downward curve or the vertices of a convex $n$-gon. The proof is also formalized in Lean 4, a computer proof assistance, to ensure the correctness of the proof.