Geodesic
Bounded fractional intersecting families are linear in size
Niranjan Balachandran1 ; Shagnik Das2 ; Brahadeesh Sankarnarayanan1
1Indian Institute of Technology Bombay
2National Taiwan University
The electronic journal of combinatorics, Tome 32 (2025) no. 3
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Using the sunflower method, we show that if $\theta \in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $\theta$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert \leq (\frac{3}{2} + o(1))n$. This partially solves a conjecture of Balachandran, Mathew and Mishra that any $\theta$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets.
DOI : 10.37236/12900
Classification : 05D05, 03E05
Mots-clés : sunflower method, intersecting families of set systems

Niranjan Balachandran  1   ; Shagnik Das  2   ; Brahadeesh Sankarnarayanan  1

1 Indian Institute of Technology Bombay
2 National Taiwan University 
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Niranjan Balachandran; Shagnik Das; Brahadeesh Sankarnarayanan. Bounded fractional intersecting families are linear in size. The electronic journal of combinatorics, Tome 32 (2025) no. 3. doi: 10.37236/12900

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