arXiv subjects

Boris Bukh

Publications and source records attributed to Boris Bukh.

At least 37 records · Page 2Linked to original sources

arXiv2021

On convex holes in $d$-dimensional point sets

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as $(t,m,s)$-nets or $(t,s)$-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of $(t,m,s)$-nets, obeying a relaxed equidistribution condition.

math.CO↗

arXiv2021

Empty axis-parallel boxes

We show that, for every set of $n$ points in the $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $Ω(d/n)$ as $n\to\infty$ and $d$ is fixed. In the opposite direction, we give a construction without an empty axis-parallel box of volume $O(d^2\log d/n)$. These improve on the previous best bounds of $Ω(\log d/n)$ and $O(2^{7d}/n)$ respectively.

math.CO↗

arXiv2020

Order-isomorphic twins in permutations

Let $a_1,\dotsc,a_n$ be a permutation of $[n]$. Two disjoint order-isomorphic subsequences are called \emph{twins}. We show that every permutation of $[n]$ contains twins of length $Ω(n^{3/5})$ improving the trivial bound of $Ω(n^{1/2})$. We also show that a random permutation contains twins of length $Ω(n^{2/3})$, which is sharp.

math.CO↗

arXiv2019

Turán numbers of theta graphs

The theta graph $Θ_{\ell,t}$ consists of two vertices joined by $t$ vertex-disjoint paths of length $\ell$ each. For fixed odd $\ell$ and large $t$, we show that the largest graph not containing $Θ_{\ell,t}$ has at most $c_{\ell} t^{1-1/\ell}n^{1+1/\ell}$ edges and that this is tight apart from the value of $c_{\ell}$.

math.CO↗

arXiv2019

On a topological version of Pach's overlap theorem

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 topological extension of Pach's theorem does not hold with subsets of size $C(\log n)^{1/(d-1)}$. We show that this is tight in dimension $2$, for all surfaces other than $\mathbb{S}^2$. Surprisingly, the optimal bound for $\mathbb{S}^2$ in the topological version of Pach's theorem is of the order $(\log n)^{1/2}$. We conjecture that, among higher-dimensional manifolds, spheres are similarly distinguished. This improves upon the results of Bárány, Meshulam, Nevo and Tancer.

math.CO↗

arXiv2019

Nearly orthogonal vectors and small antipodal spherical codes

How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $θ(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $θ(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $θ(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establish asymptotics for general $k$. The main tool is an upper bound on $\mathbb{E}_{x,y\simμ}|\langle x,y\rangle|$ whenever $μ$ is an isotropic probability mass on $\mathbb{R}^k$, which may be of independent interest. Our results translate naturally to the analogous question in $\mathbb{C}^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $\mathbb{C}^k$, also known as SIC-POVM in physics literature.

math.CO↗

arXiv2019

A bound on the number of edges in graphs without an even cycle

We show that, for each fixed $k$, an $n$-vertex graph not containing a cycle of length $2k$ has at most $80\sqrt{k}\log k\cdot n^{1+1/k}+O(n)$ edges.

math.CO↗

arXiv2019

Linear orderings of combinatorial cubes

We show that, for every linear ordering of $[2]^n$, there is a large subcube on which the ordering is lexicographic. We use this to deduce that every long sequence contains a long monotone subsequence supported on an affine cube. More generally, we prove an analogous result for linear orderings of $[k]^n$. We show that, for every such ordering, there is a large subcube on which the ordering agrees with one of approximately $\frac{(k-1)!}{2(\ln 2)^k}$ orderings.

math.CO↗

arXiv2018

On a fractional version of Haemers' bound

In this note, we present a fractional version of Haemers' bound on the Shannon capacity of a graph, which is originally due to Blasiak. This bound is a common strengthening of both Haemers' bound and the fractional chromatic number of a graph. We show that this fractional version outperforms any bound on the Shannon capacity that could be attained through Haemers' bound. We show also that this bound is multiplicative, unlike Haemers' bound.

cs.IT↗

arXiv2018

List-decodable zero-rate codes

We consider list-decoding in the zero-rate regime for two cases: the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal $τ\in [0,1]$ for which there exists an arrangement of $M$ balls of relative Hamming radius $τ$ in the binary hypercube (of arbitrary dimension) with the property that no point of the latter is covered by $L$ or more of them. As $M\to \infty$ the maximal $τ$ decreases to a well-known critical value $τ_L$. In this work, we prove several results on the rate of this convergence. For the binary case, we show that the rate is $Θ(M^{-1})$ when $L$ is even, thus extending the classical results of Plotkin and Levenshtein for $L=2$. For $L=3$ the rate is shown to be $Θ(M^{-\tfrac{2}{3}})$. For the similar question about spherical codes, we prove the rate is $Ω(M^{-1})$ and $O(M^{-\tfrac{2L}{L^2-L+2}})$.

cs.IT↗

arXiv2018

Consistent sets of lines with no colorful incidence

We consider incidences among colored sets of lines in $\mathbb{R}^d$ and examine whether the existence of certain concurrences between lines of $k$ colors force the existence of at least one concurrence between lines of $k+1$ colors. This question is relevant for problems in 3D reconstruction in computer vision.

cs.CG↗

arXiv2018

Bipartite algebraic graphs without quadrilaterals

Let $\mathbb{P}^s$ be the $s$-dimensional complex projective space, and let $X, Y$ be two non-empty open subsets of $\mathbb{P}^s$ in the Zariski topology. A hypersurface $H$ in $\mathbb{P}^s\times\mathbb{P}^s$ induces a bipartite graph $G$ as follows: the partite sets of $G$ are $X$ and $Y$, and the edge set is defined by $\overline{u}\sim\overline{v}$ if and only if $(\overline{u},\overline{v})\in H$. Motivated by the Turán problem for bipartite graphs, we say that $H\cap (X\times Y)$ is $(s,t)$-grid-free provided that $G$ contains no complete bipartite subgraph that has $s$ vertices in $X$ and $t$ vertices in $Y$. We conjecture that every $(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\overline{y}$ is bounded by a constant $d = d(s,t)$, and we discuss possible notions of the equivalence. We establish the result that if $H\cap(X\times \mathbb{P}^2)$ is $(2,2)$-grid-free, then there exists $F\in \mathbb{C}[\overline{x},\overline{y}]$ of degree $\le 2$ in $\overline{y}$ such that $H\cap(X\times \mathbb{P}^2) = \{F = 0\}\cap (X\times \mathbb{P}^2)$. Finally, we transfer the result to algebraically closed fields of large characteristic.

math.CO↗

arXiv2018

Length of the longest common subsequence between overlapping words

Given two random finite sequences from $[k]^n$ such that a prefix of the first sequence is a suffix of the second, we examine the length of their longest common subsequence. If $\ell$ is the length of the overlap, we prove that the expected length of an LCS is approximately $\max(\ell, \mathbb{E}[L_n])$, where $L_n$ is the length of an LCS between two independent random sequences. We also obtain tail bounds on this quantity.

math.PR↗

arXiv2017

Rational exponents in extremal graph theory

Given a family of graphs $\mathcal{H}$, the extremal number $\textrm{ex}(n, \mathcal{H})$ is the largest $m$ for which there exists a graph with $n$ vertices and $m$ edges containing no graph from the family $\mathcal{H}$ as a subgraph. We show that for every rational number $r$ between $1$ and $2$, there is a family of graphs $\mathcal{H}_r$ such that $\textrm{ex}(n, \mathcal{H}_r) = Θ(n^r)$. This solves a longstanding problem in the area of extremal graph theory.

math.CO↗

arXiv2017

Classifying unavoidable Tverberg partitions

Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverberg's theorem, and call a partition $\mathcal I$ of $\{1,2,\ldots,T(d,r)\}$ into $r$ parts a "Tverberg type". We say that $\mathcal I$ "occurs" in an ordered point sequence $P$ if $P$ contains a subsequence $P'$ of $T(d,r)$ points such that the partition of $P'$ that is order-isomorphic to $\mathcal I$ is a Tverberg partition. We say that $\mathcal I$ is "unavoidable" if it occurs in every sufficiently long point sequence. In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for $d\le 4$. Along the way, we study the avoidability of many other geometric predicates. Our techniques also yield a large family of $T(d,r)$-point sets for which the number of Tverberg partitions is exactly $(r-1)!^d$. This lends further support for Sierksma's conjecture on the number of Tverberg partitions.

cs.CG↗

arXiv2017

Shatter functions with polynomial growth rates

We study how a single value of the shatter function of a set system restricts its asymptotic growth. Along the way, we refute a conjecture of Bondy and Hajnal which generalizes Sauer's Lemma.

math.CO↗

arXiv2016

Ranks of matrices with few distinct entries

An $L$-matrix is a matrix whose off-diagonal entries belong to a set $L$, and whose diagonal is zero. Let $N(r,L)$ be the maximum size of a square $L$-matrix of rank at most $r$. Many applications of linear algebra in extremal combinatorics involve a bound on $N(r,L)$. We review some of these applications, and prove several new results on $N(r,L)$. In particular, we classify the sets $L$ for which $N(r,L)$ is linear, and show that if $N(r,L)$ is superlinear and $L\subset \mathbb{Z}$, then $N(r,L)$ is at least quadratic. As a by-product of the work, we asymptotically determine the maximum multiplicity of an eigenvalue $λ$ in an adjacency matrix of a digraph of a given size.

math.AC↗

arXiv2016

One-sided epsilon-approximants

Given a finite point set $P\subset\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\cap C|/|P|-|A\cap C|/|A|\leq\varepsilon$ for every convex set $C$. We show that, in contrast with the usual (two-sided) weak $\varepsilon$-approximants, for every set $P\subset \mathbb{R}^d$ there exists a one-sided weak $\varepsilon$-approximant of size bounded by a function of $\varepsilon$ and $d$.

math.CO↗