Inclusion Matrices for Rainbow Subsets

Let S be a finite set, each element of which receives a color. A rainbow t-set of S is a t-subset of S in which different elements receive different colors. Let (S t) denote the set of all rainbow t-sets of S, let (S ≤t) represent the union of (S i) for i = 0,...,t, and let 2S stand for the set of all rainbow subsets of S. The rainbow inclusion matrix WS is the 2S × 2S (0, 1) matrix whose (T , K)-entry is one if and only if T ⊆ K. We write WS t,k and WS ≤t,k for the (S t) × (S k) submatrix and the (S ≤t) × (S k) submatrix of WS, respectively, and so on. We determine the diagonal forms and the ranks of WS t,k and WS ≤t,k. We further calculate the singular values of WS t,k and construct accordingly a complete system of (0, ±1) eigenvectors for them when the numbers of elements receiving any two given colors are the same. Let DS t,k denote the integral lattice orthogonal to the rows of WS ≤t,k and let D¯S t,k denote the orthogonal lattice of DS t,k . We make use of Frankl rank to present a (0, ±1) basis of DS t,k and a (0, 1) basis of D¯S t,k. For any commutative ring R, those nonzero functions f ∈ R² S satisfying WS t,≥0 f = 0 are called null t-designs over R, while those satisfying WS ≤t,≥0 f = 0 are called null (≤ 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.