A note on the no-$(d+2)$-on-a-sphere problem

TL;DR

This work addresses the problem of selecting many lattice points from the $d$-dimensional cube $[n]^d$ with no $d+2$ points lying on a sphere or a hyperplane. It introduces a probabilistic framework augmented by VC-dimension techniques and incidence geometry to construct a large subset of size $n^{\frac{3}{d+1}-o(1)}$ free of any $d+2$-point sphere/hyperplane configurations, improving the previous bound of $Ω(n^{\frac{1}{d-1}})$. The approach blends Chernoff-type concentration, hyperplane and sphere counting bounds, a VC-dimension-based incidence bound, and a deletion argument to eliminate remaining bad tuples. The result advances understanding of high-dimensional combinatorial geometry in lattice settings and suggests potential further improvements toward $Ω(n^{\frac{d}{d+1}})$, with implications for related geometric extremal problems.

Abstract

For fixed $d\geq 3$, we construct subsets of the $d$-dimensional lattice cube $[n]^d$ of size $n^{\frac{3}{d + 1} - o(1)}$ with no $d+2$ points on a sphere or a hyperplane. This improves the previously best known bound of $Ω(n^{\frac{1}{d-1}})$ due to Thiele from 1995.

Theorems & Definitions (14)

• Theorem 1.1
• Conjecture 1.2
• Theorem 2.1: fox
• Lemma 2.2
• proof
• Lemma 3.1: Chernoff's inequality
• Lemma 3.2: balogh
• Lemma 3.3
• proof
• Lemma 3.4: sheffer, Lemma 3.2