On hierarchically closed fractional intersecting families

Niranjan Balachandran¹ ; Srimanta Bhattacharya² ; Krishn Kher³ ; Rogers Mathew⁴ ; Brahadeesh Sankarnarayanan¹

¹Department of Mathematics, Indian Institute of Technology Bombay, Mumbai
²Department of Computer Science and Engineering, Indian Institute of Technology Palakkad, Palakkad
³Department of Engineering Science cum Computer Science and Engineering, Indian Institute of Technology Hyderabad, Hyderabad
⁴Department of Computer Science and Engineering, Indian Institute of Technology Hyderabad, Hyderabad

The electronic journal of combinatorics, Tome 30 (2023) no. 4

Résumé

For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1, \dotsc, A_t \in \mathcal{F}$ there exists $\theta \in L$ such that $\lvert A_1 \cap \dotsb \cap A_t \rvert \in \{ \theta \lvert A_1 \rvert, \dotsc, \theta \lvert A_t \rvert\}$. In this paper we show that for $r \geq 3$ and $L = \{\theta\}$ any fractional $r$-closed $\theta$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $\theta = 1/2$ we have a tight upper bound of $\lfloor \frac{3n}{2} \rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism.

Détail

DOI : 10.37236/11651

Classification : 05D05, 05B99, 03E05
Mots-clés : positive proper fractions, intersecting family