An analogue of the Erdős-Stone theorem for finite geometries

Details An analogue of the Erdős-Stone theorem for finite geometries An analogue of the Erdős-Stone theorem for finite geometries

2012-03-08

arxiv.org/abs/1203.1911v1

Mathematics

Combinatorics

Scientific paper

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.

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