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Source. Remark 1.4, p. 3, with its proof under "Optimizing the error term" at the end of Section 3, pp. 14-16, including Proposition 3.8 (pp. 14-15), of A. F. Holmsen, H. N. Mojarrad, J. Pach and G. Tardos, Two extensions of the Erdős-Szekeres problem, J. Eur. Math. Soc. 22 (2020), 3981-3995, arXiv:1710.11415; read in arXiv:1710.11415v3 (3 August 2020), the edition named on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page; the supporting calculation (pp. 14-16) was read for structure only. Nothing here is independently reviewed.

Statement

With $b(n)$ the pseudo-configuration function of Theorem 1.3, Remark 1.4 (p. 3) states that a less wasteful calculation, given at the end of the paper, yields

$$b(n)\le 2^{n+\left(\frac{8\sqrt2}{3}+o(1)\right)\sqrt{n\log n}}.$$

The paper states the bound for $b(n)$; since $b(n)\ge e(n)$ (p. 3), it bounds the Erdős-Szekeres function $e(n)$ of point sets in general position in the plane in the same way. The paper writes $\log$ without a base; the proof of Theorem 1.3 bounds products of binomial coefficients by powers of $2$ with exponent $2n\log n$ (p. 14), which reads as base $2$.

Proof pointer

Proposition 3.8 (pp. 14-15) refines Theorem 2.4: for an integer $k\ge3$ and a pseudo-configuration with $|P|=N\ge 2^{(1+o(1))4k}$, either some $k$-subset in convex position has spike sizes with product at least $2^{-\frac83k^2}N^k$, or some $2k$-subset in convex position has spike sizes with product at least $2^{-\frac{40}{3}k^2-o(k^2)}N^{2k}$. With the sharper binomial estimate (3.9), $\prod_i\binom{c_i+d_i-2}{c_i-1}<(ek)^{2n}$ (p. 15), the argument of Theorem 1.3 is run in each case, and $k$ is taken to be the least even integer at least $\sqrt{n\log n}/(2\sqrt2)$ (p. 16).

Dependencies

Theorem 1.3, Theorem 2.4 and Proposition 3.8 of the same paper.

Bears on

• Problem 107: through $b(n)\ge e(n)$ this makes the constant in the error term of the upper bound on the problem's $f(n)=e(n)$ explicit. It is an upper bound only and settles no value of $f(n)$.