Related papers: Thresholds versus fractional expectation-threshold…

Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson…

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the…

We give a new proof of the Frankl-R\"odl theorem on forbidden intersections, via the probabilistic method of dependent random choice. Our method extends to codes with forbidden distances, where over large alphabets our bound is…

In this paper we look at a particular problem related to under-determined linear systems of equations with sparse solutions. $\ell_1$-minimization is a fairly successful polynomial technique that can in certain statistical scenarios find…

Given a finite set of points $S\subset\mathbb{R}^d$, a $k$-set of $S$ is a subset $A \subset S$ of size $k$ which can be strictly separated from $S \setminus A $ by a hyperplane. Similarly, a $k$-facet of a point set $S$ in general position…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

Consider a random graph G in G(n,p) and the graph property: G contains a copy of a specific graph H. (Note: H depends on n; a motivating example: H is a Hamiltonian cycle.) Let q be the minimal value for which the expected number of copies…

We give a new bound on the probability that the random sum $\xi_1 v_1 +...+ \xi_n v_n$ belongs to a ball of fixed radius, where the $\xi_i$ are iid Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an application, we prove…

Neglecting many motivating details for the Park-Pham theorem (previously known as the Kahn-Kalai conjecture), the result starts with a finite set $X$, a non-trivial upper set $\mathcal{F} \subseteq 2^X$, and a particular parameterized…

This paper establishes quantitative correlation inequalities between monotone events and structured threshold objects in both the discrete cube and Gaussian space. We prove that for any increasing balanced family, there exists a linear…

The codegree squared sum ${\rm co}_2(\cal F)$ of a family (hypergraph) $\cal F \subseteq \binom{[n]} k$ is defined to be the sum of codegrees squared $d(E)^2$ over all $E\in \binom{[n]}{k-1}$, where $d(E)=|\{F\in \cal F: E\subseteq F\}|$.…

We introduce PHFL, a probabilistic extension of higher-order fixpoint logic, which can also be regarded as a higher-order extension of probabilistic temporal logics such as PCTL and the $\mu^p$-calculus. We show that PHFL is strictly more…

Let $f(n,r,k)$ be the minimal number such that every hypergraph larger than $f(n,r,k)$ contained in $\binom{[n]}{r}$ contains a matching of size $k$, and let $g(n,r,k)$ be the minimal number such that every hypergraph larger than $g(n,r,k)$…

The families $\mathcal F_0,\ldots,\mathcal F_s$ of $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ are called cross-union if there is no choice of $F_0\in \mathcal F_0, \ldots, F_s\in \mathcal F_s$ such that $F_0\cup\ldots\cup F_s=[n]$. A…

Let $T$ be a tree space (or tree network) represented by a weighted tree with $t$ vertices, and $S$ be a set of $n$ stochastic points in $T$, each of which has a fixed location with an independent existence probability. We investigate two…

Given $1\le \ell <k$ and $\delta\ge0$, let $\textbf{PM}(k,\ell,\delta)$ be the decision problem for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs with minimum $\ell$-degree at least $\delta\binom{n-\ell}{k-\ell}$.…

A family $\mathcal{F} \subset \mathcal{P}(n)$ is $r$-wise $k$-intersecting if $|A_1 \cap \dots \cap A_r| \geq k$ for any $A_1, \dots, A_r \in \mathcal{F}$. It is easily seen that if $\mathcal{F}$ is $r$-wise $k$-intersecting for $r \geq 2$,…

We consider parametric exponential families of dimension $K$ on the real line. We study a variant of \textit{boundary crossing probabilities} coming from the multi-armed bandit literature, in the case when the real-valued distributions form…

The maximum energy of the EPR model on a weighted graph is known to be upper-bounded by the sum of the total weight and the value of maximum-weight fractional matching~(MWFM). Recently, Apte, Parekh and Sud~(APS) conjecture that the bound…

The union-closed sets conjecture (sometimes referred to as Frankl's conjecture) states that every finite, nontrivial union-closed family of sets has an element that is in at least half of its members. Although the conjecture is known to be…