---

Statement

Theorem 11.6.4 (p. 294), cited to Erdős and Szekeres (1935). There is a least function $f:\mathbb{N}\to\mathbb{N}$ such that every set of $f(n)$ points in $\mathbb{E}^2$ in general position contains the vertices of a convex $n$-gon.

Bounds recorded (p. 294). $2^{n-2}+1\le f(n)\le2^{n+4n^{4/5}}$, the lower bound cited to the same 1935 paper of Erdős and Szekeres and the upper bound to Suk (2017), which the chapter says holds for $n$ sufficiently large. The chapter also states the original Erdős–Szekeres upper bound, printed as $\binom{2n-4}{n-2+1}$.

Conjecture 11.6.5 (p. 294). The chapter asks to prove or disprove that $f(n)=2^{n-2}+1$ for $n\ge3$.

The chapter does not define general position at this point; Problem 107 states the condition as no three points on a line.

Scope

Theorem 11.6.4 and the bounds are reported from their sources, not proved in the chapter; Conjecture 11.6.5 is open as posed. The printed form of the original upper bound, with lower index $n-2+1$, is reproduced as printed and not checked against the 1935 paper here.

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of J. E. Goodman, J. O'Rourke and C. D. Tóth (eds.), Handbook of Discrete and Computational Geometry, 3rd edition, CRC Press, Boca Raton, FL, 2017; Theorem 11.6.4, the bounds and Conjecture 11.6.5 on p. 294. Pages are those printed on the edition named on the source card. Suk's bound is recorded on its own card.

Read depth. Claims checked: the theorem, the bounds and the conjecture were read on the printed page.

Bears on

• Problem 107: the conjecture is the problem's equality $f(n)=2^{n-2}+1$, posed for $n\ge3$. The chapter records the lower bound $f(n)\ge2^{n-2}+1$ and Suk's upper bound and proves neither; it leaves the equality open.