Related papers: From a $(p,2)$-Theorem to a Tight $(p,q)$-Theorem

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…

Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic…

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and…

Let ${\bf A} = (A_1, \dots, A_m)$ be an $m$-tuple of bounded linear operators acting on a Hilbert space ${\cal H}$. Their joint $(p,q)$-matricial range $\Lambda_{p,q}({\bf A})$ is the collection of $(B_1, \dots, B_m) \in {\bf M}_q^m$, where…

In terms of the Fourier-Haar coefficients, a criterion is obtained for the function $f (x_1,x_2)$ to belong to the net space $N_{\bar{p},\bar{q}}(M)$ and to the Lebesgue space $L_{\bar{p}}[0,1]^2$ with mixed metric, where…

We prove Helly-type theorems for line transversals to disjoint unit balls in $\R^{d}$. In particular, we show that a family of $n \geq 2d$ disjoint unit balls in $\R^d$ has a line transversal if, for some ordering $\prec$ of the balls, any…

Reider's Theorem on the very ampleness of adjoint linear series on a complex projective algebraic surface is extended in two new directions. First, Reider-type inequalities are shown to imply nefness of linear series of the form dH - E on…

We show that for arbitrary linearly ordered set $X$ any bounded family of (not necessarily, continuous) real valued functions on $X$ with bounded total variation does not contain independent sequences. We obtain generalized Helly's…

In this paper we introduce the family of spaces $RM(p,q)$, $1\leq p,q\leq +\infty$. They are spaces of holomorphic functions in the unit disc with average radial integrability. This family contains the classical Hardy spaces (when…

A family of $k$ point sets in $d$ dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of…

Given a family of abelian covers of $\mathbb{P}^1$ and a prime $p$ of good reduction, by considering the associated Deligne--Mostow Shimura variety, we obtain lower bounds for the Ekedahl-Oort type, and the Newton polygon, at $p$ of the…

Various theorems on convergence of general space homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring $Q$--homeomorphisms are obtained. In particular, it was established by…

Let $\pi$ be a property of pairs $(G,Z)$, where $G$ is a graph and $Z\subseteq V(G)$. In the \emph{minimum $\pi$-hitting set problem}, given an input graph $G$, we want to find a smallest set $X\subseteq V(G)$ such that $X$ intersects every…

A family f_1,...,f_n of operators on a complete metric space X is called contractive if there exists lambda < 1 such that for any x,y in X we have d(f_i(x),f_i(y)) leq lambda d(x,y) for some i. Stein conjectured that for any contractive…

A finite family $\mathcal{F}$ of $d$-dimensional convex polytopes is called $k$-neighborly if $d-k\le\textup{dim}(C\cap C')\le d-1$ for any two distinct members $C,C'\in\mathcal{F}$. In 1997, Alon initiated the study of the general function…

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in…

We study two combinatorial parameters, which we denote by f(S) and h(S), associated to an arbitrary set S \subseteq R^d, where d \in N. In the nondegenerate situation, f(S) is the largest possible number of facets of a d-dimensional…

The Lagrangian skeleton of the rational homology ball $B_{p,q}$, for $0<q<p$ coprime integers, is an immersed but not embedded Lagrangian, called a $(p,q)$-pinwheel. We show that any two embeddings of Lagrangian $(p,q)$-pinwheels in…

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…

For a given shape $S$ in the plane, one can ask what is the lowest possible density of a point set $P$ that pierces ("intersects", "hits") all translates of $S$. This is equivalent to determining the covering density of $S$ and as such is…