On maximal isolation sets in the uniform intersection matrix
share
Let A_k,t be the matrix that represents the adjacency matrix of the intersection bipartite graph of all subsets of size t of {1,2,...,k}. We give constructions of large isolation sets in A_k,t, where, for a large enough k, our constructions are the best possible. We first prove that the largest identity submatrix in A_k,t is of size k-2t+2. Then we provide constructions of isolations sets in A_k,t for any t≥ 2, as follows: * If k = 2t+r and 0 ≤ r ≤ 2t-3, there exists an isolation set of size 2r+3 = 2k-4t+3. * If k ≥ 4t-3, there exists an isolation set of size k. The construction is maximal for k≥ 4t-3, since the Boolean rank of A_k,t is k in this case. As we prove, the construction is maximal also for k = 2t, 2t+1. Finally, we consider the problem of the maximal triangular isolation submatrix of A_k,t that has ones in every entry on the main diagonal and below it, and zeros elsewhere. We give an optimal construction of such a submatrix of size (2t t-1) × (2t t-1), for any t ≥ 1 and a large enough k. This construction is tight, as there is a matching upper bound, which can be derived from a theorem of Frankl about skew matrices.
READ FULL TEXT
