On the Erd\H{o}s-Szekeres Conjecture
Hossein Nassajian Mojarrad · arXiv (Cornell University) · 2015
Let $ES(n)$ denote the minimum natural number such that every set of $ES(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, Erd\H{o}s and Szekeres proved that $ES(n) \le {2n-4 \choose n-2}+1$. 26 years later, they obtained the lower bound $2^{n-2}+1 \le ES(n)$. In a recent paper, Vlachos proved that $\limsup\limits_{n\rightarrow\infty} \frac{ES(n)}{{2n-5 \choose n-2}} \le \frac{29}{32}$. In this paper, we mostly use the ideas and tools from Vlachos' paper and improve the bound to $\limsup\limits_{n\rightarrow\infty} \frac{ES(n)}{{2n-5 \choose n-2}} \le \frac{7}{8}$.