# On Erdős–Ko–Rado for Random Hypergraphs II

@article{Hamm2018OnEF,
title={On Erdős–Ko–Rado for Random Hypergraphs II},
author={Arran Hamm and Jeff Kahn},
journal={Combinatorics, Probability and Computing},
year={2018},
volume={28},
pages={61 - 80}
}
• Published 23 June 2014
• Mathematics
• Combinatorics, Probability and Computing
Denote by ${\mathcal H}_k$(n, p) the random k-graph in which each k-subset of {1,. . .,n} is present with probability p, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed ε > 0 such that if n = 2k + 1 and p > 1 - ε, then w.h.p. (that is, with probability tending to 1 as k → ∞), ${\mathcal H}_k$(n, p) has the ‘Erdős–Ko–Rado property’. We also mention a similar random version of Sperner's theorem.
Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability
• Mathematics
Combinatorics, Probability and Computing
• 2017
The asymptotic size of the largest intersecting family in $\mathcal{H}$ k , for essentially all values of p and k, is settled.
Sharp threshold for the Erdős–Ko–Rado theorem
• Mathematics
Random Structures &amp; Algorithms
• 2022
For positive integers n and k with n > 2 k + 1, the Kneser graph K ( n, k ) is the graph with vertex set consisting of all k -sets of { 1 , . . . , n } , where two k -sets are adjacent exactly when
A sharp threshold for a random version of Sperner's Theorem
• Mathematics
• 2022
The Boolean lattice P ( n ) consists of all subsets of [ n ] = { 1 , . . . , n } partially ordered under the containment relation. Sperner’s Theorem states that the largest antichain of the Boolean
The Number of Maximal Independent Sets in the Hamming Cube
• Mathematics
Combinatorica
• 2022
Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically $2n2^{N/4},$ as was conjectured by Ilinca and the first
Introduction to Random Graphs
• Computer Science
• 2016
All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
Problems in combinatorics: Hamming cubes and thresholds
OF THE DISSERTATION PROBLEMS IN COMBINATORICS: HAMMING CUBES AND THRESHOLDS
On Erdős–Ko–Rado for random hypergraphs I
• Mathematics
Combinatorics, Probability and Computing
• 2019
The Erdős–Ko–Rado property (or is EKR) is given if each of its largest intersecting subfamilies has non-empty intersection.

## References

SHOWING 1-10 OF 44 REFERENCES
• Mathematics
Combinatorics, Probability and Computing
• 2009
It is proved that every non-trivial intersecting k-uniform hypergraph can be covered by k2 − k + 1 pairs, which is sharp as evidenced by projective planes, which improves upon a result of Sanders.
Extremal subgraphs of random graphs
• Mathematics
SODA '07
• 2007
It is proved that there is a constant c = 0, such that whenever p ≥ p, then every maximum cut of the binomial random graph G<inf>n, p, p is (<i-1)-partite, which answers a question of Babai, Simonovits and Spencer.
Threshold functions for Ramsey properties
• Mathematics
• 1995
Probabilistic methods have been used to approach many problems of Ramsey theory. In this paper we study Ramsey type questions from the point of view of random structures. Let K(n, N) be the random
Maximum Antichains in Random Subsets of a Finite Set
• D. Osthus
• Mathematics
J. Comb. Theory, Ser. A
• 2000
We consider the random poset P(n,p) which is generated by first selecting each subset of [n]={1,?,n} with probability p and then ordering the selected subsets by inclusion. We give asymptotic
A random version of Sperner's theorem
• Mathematics
J. Comb. Theory, Ser. A
• 2014
Extremal subgraphs of random graphs
• Mathematics
J. Graph Theory
• 1990
It is proved that if L is a 3-chromatic (so called “forbidden”) graph, and —Rn is a random graph on n vertices, whose edges are chosen independently, with probability p, then —Fn is an L-free subgraph of Rn of maximum size and is almost surely bipartite.
Extremal Results in Random Graphs
• Mathematics
• 2013
According to Paul Erdős au][Some notes on Turan’s mathematical work, J. Approx. Theory 29 (1980), page 4]_it was Paul Turan who “created the area of extremal problems in graph theory”. However,
On the evolution of random graphs
• Mathematics
• 1984
(n) k edges have equal probabilities to be chosen as the next one . We shall 2 study the "evolution" of such a random graph if N is increased . In this investigation we endeavour to find what is the
Large triangle-free subgraphs in graphs withoutK4
• Mathematics
Graphs Comb.
• 1986
It is shown that for arbitrary positiveε there exists a graph withoutK4 and so that all its subgraphs containing more than 1/2 +ε portion of its edges contain a triangle (Theorem 2), and it is proved that such graphs have necessarily low edge density.