The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for $n \geqslant k \geqslant t \geqslant 2$, we consider a collection of $k$ families $\mathcal{A}_i: 1 \leq i \leqslant k$ where $\mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \}$ so that $A_{1, i_1} \cap \cdots \cap A_{k,i_k} \neq \varnothing$ if and only if there are at least $t$ distinct indices $i_1,i_2,\dots,i_k$. Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size $\beta_{k,t}(n)$ of the families with ground set $[n]$.
@article{10_37236_9627,
author = {Jason O'Neill and Jacques Verstraete},
title = {A generalization of the {Bollob\'as} set pairs inequality},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {3},
doi = {10.37236/9627},
zbl = {1467.05261},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9627/}
}
TY - JOUR
AU - Jason O'Neill
AU - Jacques Verstraete
TI - A generalization of the Bollobás set pairs inequality
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/9627/
DO - 10.37236/9627
ID - 10_37236_9627
ER -
%0 Journal Article
%A Jason O'Neill
%A Jacques Verstraete
%T A generalization of the Bollobás set pairs inequality
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/9627/
%R 10.37236/9627
%F 10_37236_9627
Jason O'Neill; Jacques Verstraete. A generalization of the Bollobás set pairs inequality. The electronic journal of combinatorics, Tome 28 (2021) no. 3. doi: 10.37236/9627
