Related papers: Improved bounds on the Hadwiger-Debrunner numbers

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…

Let $q = p^e$, where $p$ is a prime and $e$ is a positive integer. The family of graphs $D(k, q)$, defined for any positive integer $k$ and prime power $q$, were introduced by Lazebnik and Ustimenko in 1995. To this day, the connected…

The topological Tverberg theorem claims that for any continuous map of the (q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images have a non-empty intersection. This has been proved for affine maps, and if $q$ is a…

Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $\Lambda_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a…

The Topological Tverberg Theorem claims that any continuous map of a (q-1)(d+1)-simplex to \R^d identifies points from q disjoint faces. (This has been proved for affine maps, for d=1, and if q is a prime power, but not yet in general.) The…

We define a reduced $\infty$-operad $\mathcal{P}$ to be $d$-connected if the spaces $\mathcal{P}\left(n\right)$, of $n$-ary operations, are $d$-connected for all $n\ge0$. Let $\mathcal{P}$ and $\mathcal{Q}$ be two reduced $\infty$-operads.…

Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least…

In this note besides two abstract versions of the Vitali Covering Lemma an abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1,1) maximal function inequality, associated to any outer measure and a family of subsets on a…

Finding a good bound on the maximal edge diameter $\Delta(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known,…

The approximate Carath\'eodory problem in general form is as follows: Given two symmetric convex bodies $P,Q \subseteq \mathbb{R}^m$, a parameter $k \in \mathbb{N}$ and $\mathbf{z} \in \textrm{conv}(X)$ with $X \subseteq P$, find…

We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1, x_2, x_3) \mapsto (s x_1, tx_2, st x_3)$ by proving necessary and sufficient conditions for the boundedness and compactness of…

Upper bounds are obtained for the $p$-capacity of compact sets in $\R^d$, with $d \ge 2$ and $1<p<d$. Upper and lower bounds are obtained for the product of $p$-capacity and powers of the $q$-torsional rigidity over the collection of all…

We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in…

A $(d,k)$-set is a subset of $\mathbb{R}^d$ containing a $k$-dimensional unit ball of all possible orientations. Using an approach of D.~Oberlin we prove various Fourier dimension estimates for compact $(d,k)$-sets. Our main interest is in…

We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $F$, the exponent $d$ is relatively…

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q = 2 case, where the representation is the mass zero, spin zero representation realized in a…

If $q = p^n$ is a prime power, then a $d$-dimensional \emph{$q$-Butson Hadamard matrix} $H$ is a $d\times d$ matrix with all entries $q$th roots of unity such that $HH^* = dI_d$. We use algebraic number theory to prove a strong constraint…

For a finite group $G,$ $\mathsf{D}(G)$ is defined as the least positive integer $k$ such that for every sequence $S=g_1\bdot g_2\bdot \dotsc \bdot g_k$ of length $k$ over $G$, there exist $1 \le i_1 < i_2 <\cdots < i_m \le k $ such that…

We study the minimum degree threshold $\delta_{r,q}$ guaranteeing the existence of $K_r$-tilings of high discrepancy in any $q$-edge-coloring. Balogh, Csaba, Pluh\'ar and Treglown handled the 2-color case, proving that $\delta_{r,2} =…

Given a fixed graph $H$, a real number $p\in(0,1)$, and an infinite Erd\H{o}s-R\'enyi graph $G\sim G(\infty,p)$, how many adjacency queries do we have to make to find a copy of $H$ inside $G$ with probability $1/2$? Determining this number…