Circulant almost cross intersecting families

Let ℱ and G be two t-uniform families of subsets over [k] = {1, 2, ..., k}, where |ℱ| |G|, let C the adjacency matrix bipartite graph whose vertices are in G, there is an edge between A ∈ B if only ∩ ≠ ∅. The pair (ℱ,G) q-almost cross intersecting every row column has exactly q zeros. We further restrict our attention to pairs that have a circulant intersection Cp, q, determined by vector with p > 0 ones followed This family matrices includes identity one extreme, crown other extreme. give constructions for wide range values parameters some cases also prove matching upper bounds. Specifically, we results following parameters: (1) 1 ≤ 2t − k + 1. (2) t2 any 0, ≥ q. (3) exponential t, large enough k. Using first result show 4t 3 then C2t 1, maximal isolation submatrix size × 1-matrix Ak, rows columns labeled all t [k], entry on x y x, intersect.