Set systems with union and intersection constraints

Set systems with union and intersection constraints

Let 2 ≤ d ≤ k be fixed and n be sufficiently large. Suppose that G is a collection of k-element subsets of an n-element set, and |G| > ( n−1 k−1 ) . Then G contains d sets with union of size at most 2k and empty intersection. This verifies a conjecture of the first author for large n.

Set Constraints with Intersection

Set constraints are inclusions between expressions denoting sets of trees. The eficiency of their satisfiabili ty test is a central issue in set-based program analysis, their main application domain. W e introduce the class of set constraints with intersection (the only operators forming the expressions are constructors and intersection) and show that i ts satisfiability problem as DEXPTIME-com...

Union-Intersecting Set Systems

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of any t sets. The maximal size of such a set system is determined exactly if s + t ≤ 4, and asymptotically if s + t ≥ 5. Finally, we exactly determine the maxi...

Set Systems with No Singleton Intersection

Let F be a k-uniform set system defined on a ground set of size n with no singleton intersection; i.e., no pair A,B ∈ F has |A ∩ B| = 1. Frankl showed that |F| ≤ (n−2 k−2 ) for k ≥ 4 and n sufficiently large, confirming a conjecture of Erdős and Sós. We determine the maximum size of F for k = 4 and all n, and also establish a stability result for general k, showing that any F with size asymptot...

Intersection and Union Logic