Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Greedy capsets

Oliver Dawson, Oleg Shuvaev, José Felipe Voloch

TL;DR

This work addresses the capset problem in ${\mathbb F}_3^n$ by examining a greedy removal procedure that starts from the full ambient space and repeatedly deletes the point with the most incident lines in the current set. Grounded in affine-space geometry, it formalizes lines via $P+Q+R=0$ and proves a structural rigidity result: greedy capsets are recursively built by removing a hyperplane $H_0$ with empty intersection and then applying the same procedure inside the two parallel hyperplanes $H_1$ and $H_2$, yielding capsets of size $|C|=2^n$ (with base case $n=1$ matching). This explains why the greedy approach does not produce larger capsets and identifies the canonical example $C=\{0,1\}^n$ as a prototypical greedy capset. The paper also discusses variants and computational observations, noting that alternative greedy strategies did not improve capset sizes.

Abstract

A capset is a subset $C \subset \mathbb{F}_3^n$ with no three points on a line. We characterise the capsets produced by successively removing points from the ambient space such that the removed point has the maximum number of lines contained in the set of remaining points and passing through it until the set of remaining points contains no lines.

Greedy capsets

TL;DR

This work addresses the capset problem in by examining a greedy removal procedure that starts from the full ambient space and repeatedly deletes the point with the most incident lines in the current set. Grounded in affine-space geometry, it formalizes lines via and proves a structural rigidity result: greedy capsets are recursively built by removing a hyperplane with empty intersection and then applying the same procedure inside the two parallel hyperplanes and , yielding capsets of size (with base case matching). This explains why the greedy approach does not produce larger capsets and identifies the canonical example as a prototypical greedy capset. The paper also discusses variants and computational observations, noting that alternative greedy strategies did not improve capset sizes.

Abstract

A capset is a subset with no three points on a line. We characterise the capsets produced by successively removing points from the ambient space such that the removed point has the maximum number of lines contained in the set of remaining points and passing through it until the set of remaining points contains no lines.

Paper Structure

This paper contains 3 sections, 3 theorems.

Table of Contents

  1. Introduction
  2. Affine spaces
  3. Greedy constructions

Key Result

Lemma 2.1

Let $S$ be a subset of the affine space $V$ that generates $V$ and assume that for any two points $P,Q \in S$ the line $\overline{PQ}$ is contained in $S$. Then $S=V$.

Theorems & Definitions (11)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Example 3.3
  • Definition 3.4
  • ...and 1 more