Handbook of combinatorics (vol. 2)
Transversal numbers for hypergraphs arising in geometry
Advances in Applied Mathematics
Multinerves and helly numbers of acyclic families
Proceedings of the twenty-eighth annual symposium on Computational geometry
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For a simplicial complex X and a field K, let h˜i(X) = dim H˜i (X; K).It is shown that if X, Y are complexes on the same vertex set, then for k ≥ 0 h˜k-1(X ∩ Y) ≤ Σσ ∈ Y Σi+j=k h˜i-1 (X[σ])ċ h˜j-1 (lk(Y, σ)).A simplicial complex X is d-Leray over K, if H˜i(Y; K) = 0 for all induced subcomplexes Y ⊂ X and i ≥ d. Let LK(X) denote the minimal d such that X is d-Leray over K. The above theorem implies that if X, Y are simplicial complexes on the same vertex set then LK(X ∩ Y) ≤ LK(X) + LK(Y).Reformulating this inequality in commutative algebra terms, we obtain the following result conjectured by Terai: If I, J are square-free monomial ideals in S = K[x1, ..., xn], then reg(I + J) ≤ reg(I) + reg(J) - 1, where reg (I) denotes the Castelnuovo-Mumford regularity of I.