Serial exchanges in random bases

Abstract

A well-known symmetric exchange property in matroid theory states that for any two bases $B_1$ and $B_2$ of a matroid and any subset $X\subseteq B_1$, there is a subset $Y\subseteq B_2$ for which $B_1-X+Y$ and $B_2-Y+X$ are bases. There have been a number of proposed strengthenings of this property. As a strengthening of a well-known conjecture of Gabow, Cordovil and Moreira, Kotlar and Ziv (2012) postulated that for any two bases $B_1$ and $B_2$ and any subset $X\subseteq B_1$, there is a subset $Y\subseteq B_2$ and orderings $x_1\prec x_2\prec \cdots \prec x_k$ and $y_1\prec y_2\prec \cdots \prec y_k$ of $X$ and $Y$, respectively, such that for $i=1,\dots ,k$, $B_1-\{x_1,\dots ,x_i\}+\{y_1,\dots ,y_i\}$ and $B_2-\{y_1,\dots ,y_i\}+\{x_1,\dots ,x_i\}$ are bases; that is, there is a subset $Y\subseteq B_2$ for which $X$ is serially exchangeable with $Y$. Progress on this problem has been very limited; to date, this conjecture has only been verified when $\left|X\right|\le 2$. In this paper, we show that for matroids representable over a finite field other than $Z_2$, the conjecture is true with high probability when $B_1$, $B_2$ and $X$ are chosen randomly, provided $\left|X\right|\le ln\left(n\right)$, where $n$ is the rank of the matroid.