Triangle-Intersecting Families of Graphs

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.Comment: 43 page

Intersecting Families of Permutations

A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.Comment: 'Erratum' section added. Yuval Filmus has recently pointed out that the 'Generalised Birkhoff theorem', Theorem 29, is false for k > 1, and so is Theorem 27 for k > 1. An alternative proof of the equality part of the Deza-Frankl conjecture is referenced, bypassing the need for Theorems 27 and 2

Hyper-regular graphs and high dimensional expanders

Let $G= (V,E)$ be a finite graph. For $d_0 > 0$ we say that $G$ is $d_0$-regular, if every $v\in V$ has degree $d_0$. We say that $G$ is $(d_0, d_1)$-regular, for $d_1>d_0>0$, if $G$ is $d_0$ regular and for every $v\in V$, the subgraph induced on $v$'s neighbors is $d_1$-regular. Similarly, $G$ is $(d_0, d_1,\ldots, d_{n-1})$-regular for $d_{n-1}>\ldots>d_1>d_0>0$, if $G$ is $d_0$ regular and for every $v\in V$, the subgraph induced on $v$'s neighbors is $(d_1,\ldots, d_{n-1})$-regular; In that case, we say that $G$ is an $n$-dimensional hyper-regular graph (HRG). Here we define a new kind of graph product, through which we build examples of infinite families of $n$-dimensional HRG such that the joint neighborhood of every clique of size at most $n-1$ is connected. In particular, relying on the work of Kaufman and Oppenheim, our product yields an infinite family of $n$-dimensional HRG for arbitrarily large $n$ with good expansion properties. This answers a question of Dinur regarding the existence of such objects.Comment: 27 page

A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\hat c=\hat c(n)$ with $0 0$, as $n$ tends to infinity Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.

Thresholds and expectation-thresholds of monotone properties with small minterms

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $\mu_p$ on the power set of $N$, where for $\mathcal{A} \subseteq 2^N$ $\mu_p(\mathcal{A}) := Pr[N_p \in \mathcal{A}]$. In this paper we study upward-closed families $\mathcal{A}$ for which all minimal sets in $\mathcal{A}$ have size at most $k$, for some positive integer $k$. We prove that for such a family $\mu_p(\mathcal{A}) / p^k$ is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables to identify in $\mathcal{A}$ either a substantial subfamily $\mathcal{A}_0$ for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than $k$. Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using duality of linear programming. This is related to the threshold conjecture of Kahn and Kalai