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Stability and exact Turan numbers for matroids by Liu, Hong; Luo, Sammy; Nelson, Peter; Nomoto, Kazuhiro is a scholarly article available to read on EtoBox.

What is Stability and exact Turan numbers for matroids about?

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 $|M|$.

Author

Liu, Hong; Luo, Sammy; Nelson, Peter; Nomoto, Kazuhiro

Published

2017

Language

EN