Circulant almost cross intersecting families
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Let ℱ and 𝒢 be two t-uniform families of subsets over [k] = {1,2,...,k}, where |ℱ| = |𝒢|, and let C be the adjacency matrix of the bipartite graph whose vertices are the subsets in ℱ and 𝒢, and there is an edge between A∈ℱ and B ∈𝒢 if and only if A ∩ B ≠∅. The pair (ℱ,𝒢) is q-almost cross intersecting if every row and column of C has exactly q zeros. We consider q-almost cross intersecting pairs that have a circulant intersection matrix C_p,q, determined by a column vector with p > 0 ones followed by q > 0 zeros. This family of matrices includes the identity matrix in one extreme, and the adjacency matrix of the bipartite crown graph in the other extreme. We give constructions of pairs (ℱ,𝒢) whose intersection matrix is C_p,q, for a wide range of values of the parameters p and q, and in some cases also prove matching upper bounds. Specifically, we prove results for the following values of the parameters: (1) 1 ≤ p ≤ 2t-1 and 1 ≤ q ≤ k-2t+1. (2) 2t ≤ p ≤ t^2 and any q> 0, where k ≥ p+q. (3) p that is exponential in t, for large enough k. Using the first result we show that if k ≥ 4t-3 then C_2t-1,k-2t+1 is a maximal isolation submatrix of size k× k in the 0,1-matrix A_k,t, whose rows and columns are labeled by all subsets of size t of [k], and there is a one in the entry on row x and column y if and only if subsets x,y intersect.
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