A note on large H-intersecting families

Nathan Keller, Noam Lifshitz

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Abstract

A family F of graphs on a fixed set of n vertices is called triangle-intersecting if for any G1,G2 ∈ F , the intersection G1 ∩ G2 contains a triangle. More generally, for a fixed graph H, a family F is H-intersecting if the intersection of any two graphs in F contains a subgraph isomorphic to H. In [D. Ellis, Y. Filmus, and E. Friedgut, J. Eur. Math. Soc., 14 (2012), pp. 841-885], a 36-year old conjecture of Simonovits and Sós was proved stating that the maximal size of a triangleintersecting family is (1/8)2n (n-1)/2. Furthermore, they proved a p-biased generalization, stating that for any p ≤ 1/2, we have μp (F ) ≤ p3, where μ p (F ) is the probability that the random graph G(n, p) belongs to \scrF . In the same paper, the authors conjectured that the assertion of their biased theorem holds also for 1/2 < p ≤ 3/4, and more generally, that for any non-t-colorable graph H and any H-intersecting family F , we have μ p (F ) ≤ pt (t+1)/2 for all p ≤ (2t 1)/(2t). In this note we construct, for any fixed H and any p > 1/2, an H-intersecting family F of graphs such that μ p (F ) ≥ 1 e-n2/C, where C depends only on H and p, thus disproving both conjectures.

Original languageAmerican English
Pages (from-to)398-401
Number of pages4
JournalSIAM Journal on Discrete Mathematics
Volume33
Issue number1
DOIs
StatePublished - 2019
Externally publishedYes

Bibliographical note

Funding Information:
\ast Received by the editors October 15, 2018; accepted for publication (in revised form) December 14, 2018; published electronically March 5, 2019. http://www.siam.org/journals/sidma/33-1/M122076.html Funding: The first author's research was supported by the Israel Science Foundation (grants 402/13 and 1612/17), the Binational US-Israel Science Foundation (grant 2014290), and by the Alon Fellowship. \dagger Department of Mathematics, Bar Ilan University, Ramat Gan, 5290002, Israel (nathan.keller@ biu.ac.il, noamlifshitz@gmail.com).

Funding Information:
The first author's research was supported by the Israel Science Foundation (grants 402/13 and 1612/17), the Binational US-Israel Science Foundation (grant 2014290), and by the Alon Fellowship.

Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics.

Keywords

  • Extremal set theory
  • Intersecting families
  • Probabilistic methods
  • Triangle intersecting

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