Saturation results around the Erdős--Szekeres problem

TL;DR

The paper investigates saturation variants of the Erdős–Szekeres problems across geometry and ordered hypergraphs. It proves a tight, structure-preserving link between saturation numbers and Ramsey numbers in several settings, notably showing sat_p^{(r)}(k,ℓ)=ram_p^{(r)}(k,ℓ) for all r≥2, and establishing a strong upper bound sat_g(n) ≤ (7/8)·2^{n-2} for n≥7 in the convex-polygon case. The convex-polygon result is achieved by modifying Erdős–Szekeres' classic construction to replace substructures with smaller saturated blocks, and the same philosophy is extended to cups-versus-caps with detailed bounds and exact values. A warm-up and a general hypergraph labeling approach underpin the hypergraph results, connecting monotone paths to Ramsey theory via ordered structures. The work highlights significant gaps between saturation and Ramsey numbers in geometric settings and raises open questions about asymptotics for cups-versus-caps and holes in planar point sets.

Abstract

In this paper, we consider saturation problems related to the celebrated Erdős--Szekeres convex polygon problem. For each $n \ge 7$, we construct a planar point set of size $(7/8) \cdot 2^{n-2}$ which is saturated for convex $n$-gons. That is, the set contains no $n$ points in convex position while the addition of any new point creates such a configuration. This demonstrates that the saturation number is smaller than the Ramsey number for the Erdős--Szekeres problem. The proof also shows that the original Erdős--Szekeres construction is indeed saturated. Our construction is based on a similar improvement for the saturation version of the cups-versus-caps theorem. Moreover, we consider the generalization of the cups-versus-caps theorem to monotone paths in ordered hypergraphs. In contrast to the geometric setting, we show that this abstract saturation number is always equal to the corresponding Ramsey number.

Figures (8)

• Figure 1: Extending $H$ and its coloring over a new vertex with the label (1,3).
• Figure 2: Configurations that force a red edge by transitivity.
• Figure 3: Transitivity problems come in pairs.
• Figure 4: Around the triangle $\Delta$ spanned by $p_2,p_3,p_4$. The line $p_6p_8$ does not divide $\Delta_4$.
• Figure 5: Three different ways of combining cups and caps.

Theorems & Definitions (32)

• Theorem 1: Damásdi et al. DKMTWZ2021
• Theorem 2
• Conjecture 3
• Theorem 4
• Theorem 5
• proof : Proof of \ref{['thm:sat_p']} assuming $r = 2$
• proof : New proof of \ref{['thm:sat_s']}
• Theorem 8
• proof : Proof of \ref{['hypergraph']}
• Proposition 9