Inclusion Matrices for Rainbow Subsets

Abstract

Let \(\text {S}\) be a finite set, each element of which receives a color. A rainbow t-set of \(\text {S}\) is a t-subset of \(\text {S}\) in which different elements receive different colors. Let \(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \) denote the set of all rainbow t-sets of \(\text {S}\), let \(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \) represent the union of \(\left( {\begin{array}{c}\text {S}\\ i\end{array}}\right) \) for \(i=0,\ldots , t\), and let \(2^\text {S}\) stand for the set of all rainbow subsets of \(\text {S}\). The rainbow inclusion matrix \(\mathcal {W}^{\text {S}}\) is the \(2^\text {S}\times 2^{\text {S}}\) (0, 1) matrix whose (T, K)-entry is one if and only if \(T\subseteq K\). We write \(\mathcal {W}_{t,k}^{\text {S}}\) and \(\mathcal {W}_{\le t,k}^{\text {S}}\) for the \(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) submatrix and the \(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) submatrix of \(\mathcal {W}^{\text {S}}\), respectively, and so on. We determine the diagonal forms and the ranks of \(\mathcal {W}_{t,k}^{\text {S}}\) and \(\mathcal {W}_{\le t,k}^{\text {S}}\). We further calculate the singular values of \(\mathcal {W}_{t,k}^{\text {S}}\) and construct accordingly a complete system of \((0,\pm 1)\) eigenvectors for them when the numbers of elements receiving any two given colors are the same. Let \(\mathcal {D}^{\text {S}}_{t,k}\) denote the integral lattice orthogonal to the rows of \(\mathcal {W}_{\le t,k}^{\text {S}}\) and let \(\overline{\mathcal {D}}^{\text {S}}_{t,k}\) denote the orthogonal lattice of \(\mathcal {D}^{\text {S}}_{t,k}\). We make use of Frankl rank to present a \((0,\pm 1)\) basis of \(\mathcal {D}^{\text {S}}_{t,k}\) and a (0, 1) basis of \(\overline{\mathcal {D}}^{\text {S}}_{t,k}\). For any commutative ring R, those nonzero functions \(f\in R^{2^{\text {S}}}\) satisfying \(\mathcal {W}_{t,\ge 0}^{\text {S}}f=0\) are called null t-designs over R, while those satisfying \(\mathcal {W}_{\le t,\ge 0}^{\text {S}}f=0\) are called null \((\le t)\)-designs over R. We report some observations on the distributions of the support sizes of null designs as well as the structure of null designs with extremal support sizes.