A Meshalkin theorem for projective geometries (Q1395829)

A projective geometry \(\mathbb{P}^{n-1}(q)\) of order \(q\) and rank \(n\) (i.e. dimension \(n-1\)) is viewed as a lattice of flats in which \(\widehat 0=\emptyset\) and \(\widehat 1\) is the set of all points. A Meshalkin sequence of length \(p\) in \(\mathbb{P}^{n-1}(q)\) is a sequence \(a=(a_1,a_2,\dots,a_p)\) of flats whose join is \(\widehat 1\), and whose ranks sum to \(n\). The authors prove upper bounds on the cardinality of a family, \({\mathcal M}\), of Meshalkin sequences, and (from the authors' abstract) ``corresponding LYM inequalities, assuming that (i) all joins are the whole geometry, and, for each \(k<p\), the set of all \(a_k\)'s of sequences in \({\mathcal M}\) contains no chain of length \(\ell\), and that (ii) the joins are arbitrary, and the chain condition holds for all \(k\). These results are \(q\)-analogs of generalizations of \textit{L. D. Meshalkin}'s [Theor. Probab. Appl. 8, 203-204 (1963); translation from Teor. Veroyatn. Primen. 8, 219-220 (1963; Zbl 0123.36303)] and \textit{P. Erdős}'s [Bull. Am. Math. Soc. 51, 898-902 (1945; Zbl 0063.01270)] generalizations of Sperner's theorem and their LYM companions [\textit{M. Hochberg} and \textit{W. M. Hirsch}, Ann. N.Y. Acad. Sci. 175, 224-237 (1970; Zbl 0231.05007)], and they generalize \textit{G.-C. Rota} and \textit{L. H. Harper}'s \(q\)-analog [Matching theory, an introduction, Advances Probab. related Topics 1, 169-215 (1971; Zbl 0234.05001)] of Erdős's generalization''.