Stability and exact Turan numbers for matroids

TL;DR

This paper investigates the maximum size of subsets in binary vector spaces avoiding certain linear configurations, providing stability results and exact bounds that refine previous approximate theorems.

Contribution

It establishes a stability version of the Erdős-Stone type theorem for matroids and determines exact Turán numbers by linking error terms to solutions of sparse extremal problems.

Findings

Stability result showing sets close in size are also close in structure.

Exact Turán numbers obtained for many cases, eliminating error terms.

Connection between error bounds and solutions to sparse extremal problems.

Abstract

We consider the Tur\'an-type problem of bounding the size of a set $M \subseteq \mathbb{F}_2^n$ that does not contain a linear copy of a given fixed set $N \subseteq \mathbb{F}_2^k$, where $n$ is large compared to $k$. An Erd\H{o}s-Stone type theorem [5] in this setting gives a bound that is tight up to a $o(2^n)$ error term; our first main result gives a stability version of this theorem, showing that such an $M$ that is close in size to the upper bound in [5] is close in edit distance to the obvious extremal example. Our second result shows that the error term in [5] is exactly controlled by the solution to one of a class of `sparse' extremal problems, and in many cases eliminates the error term completely to give a sharp upper bound on $|M|$.