Related papers: An Improved Trickle-Down Theorem for Partite Compl…

This is part 2 of a 3-part article where we provide an $O(n^2)$-algorithm to produce a surgery presentation of a 3-manifold induced by a gem with a resolution. In this part we produce a sequence of colored simplicial 2-complexes which are…

This thesis addresses the question of the maximal number of $d$-simplices for a simplicial complex which is embeddable into $\mathbb{R}^r$ for some $d \leq r \leq 2d$. A lower bound of $f_d(C_{r + 1}(n)) =…

A digraph $D$ is $k$-linked if for every $2k$ distinct vertices $ x_1,\ldots , x_k, y_1, \ldots , y_k$ in $D$, there exist $k$ pairwise vertex-disjoint paths $P_1,\ldots, P_k$ such that $P_i$ starts at $x_i$ and ends at $y_i$ for each $i\in…

The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$,…

Let $\mathscr{X}$ be the boundary complex of a $(d+1)$-polytope, and let $\rho(d+1,k) = \frac{1}{2}[{\lceil (d+1)/2 \rceil \choose d-k} + {\lfloor (d+1)/2 \rfloor \choose d-k}]$. Recently, the author, answering B\'ar\'any's question from…

Fix positive integers $k$ and $d$. We show that, as $n\to\infty$, any set system $\mathcal{A} \subset 2^{[n]}$ for which the VC dimension of $\{ \triangle_{i=1}^k S_i \mid S_i \in \mathcal{A}\}$ is at most $d$ has size at most…

Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in…

A graph is $k$-gap-planar if it has a drawing in the plane such that every crossing can be charged to one of the two edges involved so that at most $k$ crossings are charged to each edge. We show this class of graphs has linear expansion.…

We study the problem of testing $k$-block-positivity via symmetric $N$-extendibility by taking the tensor product with a $k$-dimensional maximally entangled state. We exploit the unitary symmetry of the maximally entangled state to reduce…

The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let…

We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in…

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < \alpha < 1$. Let…

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a…

Various threshold effects are investigated on a discrete quasi-1D scattering system. In particular, one of these effects is to add corrections to Levinson's theorem. We explain how these corrections are due to the opening or to the closing…

Given a finite point set $P$ in ${\mathbb R}^d$, and $\epsilon>0$ we say that $N\subseteq{ \mathbb R}^d$ is a weak $\epsilon$-net if it pierces every convex set $K$ with $|K\cap P|\geq \epsilon |P|$. We show that for any finite point set in…

Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's…

We prove the following theorem, which is related to McMullen's problem on projective transformations of polytopes; let $2\leq k\leq \lfloor{\frac{d}{2}}\rfloor$ and $\nu{(d, k)}$ be the largest number such that any set of $\nu{(d,k)}$…

We prove upper bounds on the face numbers of simplicial complexes in terms on their girths, in analogy with the Moore bound from graph theory. Our definition of girth generalizes the usual definition for graphs.

Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular…

Under mild assumptions, we characterise modules with projective resolutions of length n in the target category of filtrated K-theory over a finite topological space in terms of two conditions involving certain Tor-groups. We show that the…