On the lines-planes inequality for matroids
Let \(G\) be a combinatorial geometry (or simple matroid). Let \(W_i\) be the number of rank-\(i\) flats in \(G\), so \(W_1\) is the number of points, \(W_2\) the number of lines, and so on. Rota's famous unimodality conjecture states that if \(G\) is a rank-\(n\) geometry, then there is a rank \(s\) such that \(W_0\leq W_1 \leq W_2 \leq \cdots \leq W_s\) and \(W_{s+1}\geq W_{s+2}\geq \cdots \geq W_n\). The author proves a partial result concerning the case \(n=5\). Namely, it is shown here that if \(G\) is a geometry of rank at least 5 in which all the lines have the same number of points, then \(W_2 \leq W_3\). The basic idea is to use a special case in a result of \textit{T. A. Dowling} and \textit{R. M. Wilson} [Proc. Am. Math. Soc. 47, 504-512 (1975; Zbl 0297.05010)] together with an explicit ``reconstruction'' algorithm developed by the author in [Math Proc. Camb. Philos. Soc. 101, 221-231 (1987; Zbl 0626.06008)].
scientific article; zbMATH DE number 1525314