An analogue of the Erdős-Stone theorem for finite geometries

Abstract

For a set G of points in P G(m − 1, q), let ex q (G; n) denote the maximum size of a collection of points in P G(n − 1, q) not containing a copy of G, up to projective equivalence. We show thatwhere c is the smallest integer such that there is a rank-(m − c) flat in P G(m−1, q) that is disjoint from G. The result is an elementary application of the density version of the Hales-Jewett Theorem.

The structure of claw-free binary matroids

“…This lemma will not be needed in the rest of the paper. Recall that a partial lift-join of 5 gives a contradiction unless s ≤ 2; this implies the claimed bound.…”

“…It seems that this is part of a larger phenomenon, and that our theorem forms part of the natural 'exponential' analogue of the theory of the subgraph order. Binary matroids with the 'restriction' order, as we have defined them, resemble simple graphs with the subgraph and induced subgraph order in a variety of deep contexts (see [1,2,4,5,6,8], for example).…”

Excluding affine configurations over a finite field

“…In [GN15], Geelen and Nelson proved a result in extremal matroid theory that is an analogue of the Erdős-Stone theorem for graphs. The theorem states that for a fixed set N of point in the projective space PG(m − 1, q), the maximum size ex q (N, n) of a subset A ⊆ PG(n − 1, q) not containing a copy of N satisfies exq(N,n)…”

Stability and exact Turán numbers for matroids

“…For each matroid N of critical number k, the matroid BB(n−1, 2, k −1) is N-free; the following analogue of the Erdős-Stone theorem shows that it is the largest N-free matroid up to an error term. Theorem 1.2 (Matroidal Erdős-Stone Theorem [5]). For every N,…”