Related papers: Discrete Helly-type theorems for pseudohalfplanes

Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1},…

We use methods from spectral graph theory to obtain bounds on the number of incidences between $k$-planes and $h$-planes in $\mathbb{F}_q^d$ which generalize a recent result given by Bennett, Iosevich, and Pakianathan (2014). More…

Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in…

Let $(P,E)$ be a $(d+1)$-uniform geometric hypergraph, where $P$ is an $n$-point set in general position in $\mathbb{R}^d$ and $E\subseteq {P\choose d+1}$ is a collection of $\epsilon{n\choose d+1}$ $d$-dimensional simplices with vertices…

Let $F$ be a totally real field and $p$ be an odd prime which splits completely in $F$. We prove that the eigenvariety associated to a definite quaternion algebra over $F$ satisfies the following property: over a boundary annulus of the…

The {\it Weierstrass semigroup} of pole orders of meromorphic functions in a point $p$ of a smooth algebraic curve $C$ is a classical object of study; a celebrated problem of Hurwitz is to characterize which semigroups ${\rm S} \subset…

A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is `pseudorandom', meaning that, for instance,…

A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet…

For a finite set $S$ of points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any odd set $S$ there exists a Hamiltonian cycle for which these disks share a point, and…

It is shown that a flat subgroup, $H$, of the totally disconnected, locally compact group $G$ decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, $P$, of a multiplicative semigroup in…

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of…

We show that for all $\ell, k, n$ with $\ell \leq k/2$ and $(k-\ell)$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph $\mathcal{K}_n^{(k)}$ on $n$…

Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal…

We establish second main theorems for holomorphic curves into a projective subvary $V \subset \mathbb{P}^n(\mathbb{C})$ of dimension $k$, intersecting hypersurfaces in $N$-subgeneral position with respect to $V$ $(N > k)$. Our results…

A sunflower is a collection of sets $\{U_1,\ldots, U_n\}$ such that the pairwise intersection $U_i\cap U_j$ is the same for all choices of distinct $i$ and $j$. We study sunflowers of convex open sets in $\mathbb R^d$, and provide a…

We consider collections of hyperplanes in $\mathbb{R}^n$ covering all vertices of the $n$-dimensional hypercube $\{0,1\}^n$, which satisfy the following nondegeneracy condition: For every $v\in \{0,1\}^n$ and every $i=1,\dots,n$, we demand…

We prove that the complement of a very generic curve of degree $d$ at least equal to 15 in the projective plane is hyperbolic in the sens of Kobayashi (here, the terminology ``very generic'' refers to complements of countable unions of…

Let $M$ be a compact hypersurface with boundary $\partial M=\partial D_1 \cup \partial D_2$, $\partial D_1 \subset \Pi _1$, $\partial D_2 \subset \Pi _2$, $\Pi_1$ and $\Pi _2$ two parallel hyperplanes in $\mathbb{R}^{n+1}$ ($n \geq 2$).…