Related papers: On a problem of M. Talagrand

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq…

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. Let $f(n,a)$ be the maximum number of sets in a…

In this article, we consider the family of functions $f$ meromorphic in the unit disk $\ID=\{z :\,|z| < 1\}$ with a pole at the point $z=p$, a Taylor expansion \[f(z)= z+\sum_{k=2}^{\infty} a_kz^k, \quad |z|<p, \] and satisfying the…

For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose…

Mubayi's Conjecture states that if $\mathcal{F}$ is a family of $k$-sized subsets of $[n] = \{1,\ldots,n\}$ which, for $k \geq d \geq 2$, satisfies $A_1 \cap\cdots\cap A_d \neq \emptyset$ whenever $|A_1 \cup\cdots\cup A_d| \leq 2k$ for all…

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…

Let $k\geqslant 3$ and let $A=\{0=a_{0}<a_{1}<\cdots<a_{k-1}\}$ with $\gcd(A)=1$. Freiman-Lev conjecture [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40] is a well-known…

Let $t$ be a non-negative integer and $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ be a set-pair family satisfying $|A_i \cap B_i|\leq t$ for $1\leq i \leq m$. $\mbox{$\cal P$}$ is called strong Bollob\'as $t$-system, if $|A_i\cap…

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…

Using standard methods (due to Janson, Stein-Chen, and Talagrand) from probabilistic combinatorics, we explore the following general theme: As one progresses from each member of a family of objects ${\cal A}$ being "covered" by at most one…

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…

We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called supratopological spaces. We prove some results on the…

The union-closed sets conjecture states that in any nonempty union-closed family $\mathcal{F}$ of subsets of a finite set, there exists an element contained in at least a proportion $1/2$ of the sets of $\mathcal{F}$. Using the…

Frankl and F\"uredi conjectured in 1989 that the maximum Lagrangian of all $r$-uniform hypergraphs of fixed size $m$ is realized by the minimum hypergraph $C_{r,m}$ under the colexicographic order. In this paper, we prove a weaker version…

Fra\"iss\'e's conjecture (proved by Laver) is implied by the $\Pi^1_1$-comprehension axiom of reverse mathematics, as shown by Montalb\'an. The implication must be strict for reasons of quantifier complexity, but it seems that no better…

Let $K=\{k_1,k_2,\ldots,k_r\}$ and $L=\{l_1,l_2,\ldots,l_s\}$ be disjoint subsets of $\{0,1,\ldots,p-1\}$, where $p$ is a prime and $A=\{A_1,A_2,\ldots,A_m\}$ be a family of subsets of $[n]$ such that $|A_i|\pmod{p}\in K$ for all $A_i\in A$…

We propose a counting dimension for subsets of Z and prove that, under certain conditions on two such subsets E and F, for Lebesgue almost every real \lambda\ the counting dimension of E+[\lambda F] is at least the minimum between 1 and the…

The $p$-adic Littlewood Conjecture due to De Mathan and Teuli\'e asserts that for any prime number $p$ and any real number $\alpha$, the equation $$\inf_{|m|\ge 1} |m|\cdot |m|_p\cdot |\langle m\alpha \rangle|\, =\, 0 $$ holds. Here, $|m|$…

Fuglede's conjecture states that for a subset $\Omega$ of a locally compact abelian group $G$ with positive and finite Haar measure, there exists a subset of the dual group of $G$ which is an orthogonal basis of $L^{2}(\Omega)$ if and only…