An analogue of the Erd\H{o}s-Stone theorem for finite geometries

TL;DR

This paper establishes an analogue of the Erdős-Stone theorem for finite projective geometries, determining the asymptotic maximum size of point sets avoiding a given configuration.

Contribution

It introduces a finite geometric analogue of the Erdős-Stone theorem, linking extremal set sizes to geometric configurations and using the density Hales-Jewett theorem.

Findings

Asymptotic ratio of maximum point sets avoiding G is 1 - q^{1-c}.

Identifies the minimal flat disjoint from G determines the extremal size.

Uses elementary density Hales-Jewett theorem for proof.

Abstract

For a set $G$ of points in $\PG(m-1,q)$, let $\ex_q(G;n)$, denote the maximum size of a collection of points in $\PG(n-1,q)$ not containing a copy of $G$, up to projective equivalence. We show that \[\lim_{n\rightarrow \infty} \frac{\ex_q(G;n)}{|\PG(n-1,q)|} = 1-q^{1-c},\] where $c$ is the smallest integer such that there is a rank-$(m-c)$ flat in $\PG(m-1,q)$ that is disjoint from $G$. The result is an elementary application of the density version of the Hales-Jewett Theorem.