Erdős-Ko-Rado-type results over \(J_q(n,d),H_q(n,d)\) and their designs
type: Item
instance of: scholarly article
title: Erdős-Ko-Rado-type results over \(J_q(n,d),H_q(n,d)\) and their designs
zbMATH Open document ID: 0924.05066
DOI: 10.1016/S0012-365X(98)00171-X
author: Tung-Shan Fu
published in: Discrete Mathematics
publication date: 1999-09-02T00:00:00Z
review text: Let \(\mathcal F\) be a collection of \(k\)-subsets of an \(n\)-set \(X\) satisfying the property that \(| x\cap y| \geq r\) for all \(x,y\in \mathcal F\), where \(0\leq r\leq k\). Erdős et al. proved that \(| {\mathcal F}| \leq {n-r\choose k-r}\) if \(n\geq (r+1)(k-r+1)\), and equality holds if and only if \(\mathcal F\) consists of all \(k\)-subsets which contain a fixed \(r\)-subset of \(X\). In this paper the results are proved for regular quantum matroids. A quantum matroid is any nonempty ranked poset \(\mathcal P\) satisfying the following conditions: (1) \(\mathcal P\) is a meet semilattice. (2) For all \(x\in \mathcal P\), the interval \([0,x]\) is a modular atomic lattice. (3) For all \(x,y\in {\mathcal P}\) satisfying \(\text{rank} (x)<\text{rank} (y)\), there exists an atom \(a\in {\mathcal P}\) such that \(a\leq y\), \(a\not\leq x\), and \(a\vee x\) exists in \(\mathcal P\). For a quantum matroid \(\mathcal P\), the rank of \(\mathcal P\), \(\text{rank} ({\mathcal P})\), is \(\max \{\text{rank} (x)\mid x\in {\mathcal P}\}\). The quantum matroid \(\mathcal P\) is occasionally written as \((A_0,A_1,\dots,A_d)\), where \(A_i=\{x\in {\mathcal P} \mid \text{rank} (x)=i\}\) for \(0\leq i\leq d=\text{rank} ({\mathcal P})\). For any subset \(A\subseteq \mathcal P\) and any element \(u\in \mathcal P\), we define \(A(u)=\{x\in A\mid x\leq u\) or \(u\leq x\}\). A regular quantum matroid \({\mathcal P}=(A_0,A_1,\dots,A_d)\) with parameters \((d,q,\alpha,\beta)\) is a quantum matroid of rank \(d\) satisfying the following conditions: (4) \(| A_1(u)| =q+1\) for all \(u\in A_2\). (5) For any pair of elements \((x,y)\) of \(\mathcal P\) such that \(x\in A_{d-1}\), \(y\in A_d\), and \(x\wedge y\in A_{d-2}\), \(| \{(u,v)\mid u\in A_{d}, v\in A_{d-1}, u=x\vee v, v=u\wedge y\}| = \alpha+1\). (6) \(| A_d(x)| =\beta+1\) for all \(x\in A_{d-1}\). Theorem 3.1. Let \({\mathcal P}=(A_0,A_1,\dots,A_d)\) be a regular quantum matroid with parameters \((d,q,\alpha,\beta)\), and assu…
Mathematics Subject Classification ID: 05D05
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1321771
zbMATH Keywords: association schemes
zbMATH Keywords: Delsarte \(t\)-schemes
zbMATH Keywords: regular quantum matroid
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1321771