On the Turán number of triple systems
Journal of Combinatorial Theory Series A
Extremal Graph Theory
Generating all sets with bounded unions
Combinatorics, Probability and Computing
Generating all subsets of a finite set with disjoint unions
Journal of Combinatorial Theory Series A
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Let $\cal{B}(n, \leq 4)$ denote the subsets of $[n]:=\{ 1, 2, \dots, n\}$ of at most 4 elements. Suppose that $\cal{F}$ is a set system with the property that every member of $\cal{B}$ can be written as a union of (at most) two members of $\cal{F}$. (Such an $\cal{F}$ is called a 2-base of $\cal{B}$.) Here we answer a question of Erdös proving that \[|\FF|\geq 1+n+\binom{n}{2}- \Bigl\lfloor \frac{4}{3}n\Bigr\rfloor\], and this bound is best possible for $n\geq 8$.