combinatorics / Matroid theory
Mason conjectured the following: let $M$ be a matroid of rank $r$, and let $W_i$ denote the number of flats of $M$ of rank $i$. Is it true that for all $1 \leq i \leq r - 1$, we have $W_i^2 \geq W_{i + 1}W_{i - 1}$? This is false; a counterexample is given by a graphic matroid whose graph is a generalized theta graph with $79$ edges.
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Append-only history
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
Research memory
Mason conjectured the following: let $M$ be a matroid of rank $r$, and let $W_i$ denote the number of flats of $M$ of rank $i$. Is it true that for all $1 \leq i \leq r - 1$, we have $W_i^2 \geq W_{i + 1}W_{i - 1}$? This is false; a counterexample is given by a graphic matroid whose graph is a generalized theta graph with $79$ edges.
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
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