## 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)2^{n} (n-1)/2. Furthermore, they proved a p-biased generalization, stating that for any p ≤ 1/2, we have μ_{p} (F ) ≤ p^{3}, 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 ) ≤ p^{t} (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 language | American English |
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Pages (from-to) | 398-401 |

Number of pages | 4 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |

Externally published | Yes |

### 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