On the Erd\H{o}s-Tuza-Valtr Conjecture

The Erd\H{o}s-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. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that any set of more than $\sum_{i = n - b}^{a - 2} \binom{n - 2}{i}$ 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 Erd\H{o}s-Szekeres conjecture. We prove the first new case of the Erd\H{o}s-Tuza-Valtr conjecture since the original 1935 paper of Erd\H{o}s and Szekeres. Namely, we show that any set of $\binom{n-1}{2} + 2$ 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.Comment: 16 pages, 8 figure