Abstract
Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a question of Frankl, we present some structural results for families that are $(d,s)$-conditionally intersecting with $s\ge 2k+d-3$, and families that are $(k,2k)$-conditionally intersecting. As applications of our structural results we present some new proofs to the upper bounds for the size of the following $k$-uniform families on $[n]$:
 (a) $(d,2k+d-3)$-conditionally intersecting families with $n\ge 3k^5$;
 (b) $(k,2k)$-conditionally intersecting families with $n\ge k^2/(k-1)$;
 (c) Nonintersecting $(3,2k)$-conditionally intersecting families with $n\ge 3k\binom{2k}{k}$.
 Our results for $(c)$ confirms a conjecture of Mammoliti and Britz for the case $d=3$.
Highlights
Let V be a set, and let S, T be two subsets of V
A d-cluster of k-sets is a collection of d different k-subsets A1, . . . , Ad of [n]
The following result of Bollobas [1] gives an upper bound for the size of a family in which every set has a unique subset
Summary
Let V be a set, and let S, T be two subsets of V. The following result of Bollobas [1] gives an upper bound for the size of a family in which every set has a unique subset. Frankl [6] proved the following structural result for (3, 6)-conditionally intersecting families. F can be partitioned into two families H and B, and the ground set [n] can be partitioned into two disjoint subsets Y and Z such that the following statements hold.
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