Erdős-Ko-Rado-type results over \(J_q(n,d),H_q(n,d)\) and their designs
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 assume \(d\geq 4\), \(q\geq 2\), \(\alpha \geq 1\). Further assume \(\beta\geq \alpha \sum_{i=0}^{d}q^i\) if \(q\geq 3\) and \(\beta\geq \alpha \sum_{i=0}^{d+1}q^i\) if \(q=2\). If \({\mathcal F}\subseteq A_d\) has the property that \(\text{rank} (x\wedge y)\geq r\) for all \(x,y\in \mathcal F\) (\(r<d\)), then (1) \(| {\mathcal F}| \leq \Pi_{i=r}^{d-1}((1+\beta q^{d-i-1})/e_{d-i-1})\), and (2) equality holds if and only if \({\mathcal F}=A_d(u)\) for some \(u\in A_r\). Here \(e_j=1+\alpha \sum_{i=0}^{j-1}q^i\). This theorem covers Erdős-Ko-Rado-type results for vector spaces and for bilinear forms.
scientific article; zbMATH DE number 1321771