arXiv subjects

Boris Bukh

Publications and source records attributed to Boris Bukh.

At least 19 recordsLinked to original sources

arXiv2026

Faster random walks via infrequent steering

Random walks on graphs can be slow. To speed them up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon>0$ that we can choose it. We show that in this case, at least for graphs of bounded degree, there is a way to steer the walk so that it visits every vertex in $n^{1+o(1)}$ steps with high probability. The key to this result is a way to decompose arbitrary graphs into small-diameter pieces.

math.PR↗

arXiv2025

Covering large-dimensional Euclidean spaces by random translates of a given convex body

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few and Rogers [Mathematika 6, 1959] and the upper bound of $\left(1/2+o(1) \right)n \ln n$ by Dumer [Discrete Comput. Geom. 38, 2007]. We prove that there are ball coverings of $\mathbb{R}^n$ attaining the asymptotically best known density $\left(1/2+o(1) \right)n \ln n$ such that, additionally, every point of $\mathbb{R}^n$ is covered at most $\left(1.79556... + o(1)\right) n \ln n$ times. This strengthens the result of Erd\H{o}s and Rogers [Acta Arith. 7, 1961/62] who had the maximum multiplicity at most $\left(\mathrm{e} + o(1)\right) n \ln n$. On the other hand, we show that the method that was used for the best known ball coverings (when one takes a random subset of centres in a fundamental domain of a suitable lattice in $\mathbb{R}^n$ and extends this periodically) fails to work if the density is less than $(1/2+o(1))n\ln n$; in fact, this result remains true if we replace the ball by any convex body $K$. Also, we observe that a ``worst'' convex body $K$ here is a cube, for which the packing density coming from random constructions is only $(1+o(1))n\ln n$.

math.CO↗

arXiv2025

Most frequent subsequences in a word

We prove that every $n$-letter word over $k$-letter alphabet contains some word as a subsequence in at least $k^{n/4k(1+o(1))}$ many ways, and that this is sharp as $k\to\infty$. For fixed $k$, we show that the analogous number deviates from $\mu_k^n$, for some constant $\mu_k$, by a factor of at most $n$.

math.CO↗

arXiv2025

The Oddtown problem modulo a composite number

A family of subsets $\mathcal{A}$ of an $n$-element set is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. Berlekamp and Graver showed that when $\ell$ is a prime, the maximum size of an $\ell$-Oddtown is $n$. Babai and Frankl extended this to prime powers, and asked whether the maximum size is still $n$ when $\ell$ is not a prime power, a question that was open even for $\ell=6$. For square-free composite moduli with $\omega$ distinct prime factors, the argument of Szegedy gives an upper bound of $\omega n-\omega\log_2 n$ on the size of an $\ell$-Oddtown. We answer the question of Babai and Frankl in the negative by constructing $\ell$-Oddtowns of size $\omega n-o(n)$, which shows that the leading term $\omega n$ cannot be improved. We also improve Szegedy's upper bound to $\omega n-(2\omega +\varepsilon)\log_2 n$ for most $\ell$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments.

math.CO↗

arXiv2025

Maximal sets of a given diameter in Hamming cubes

A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or contains a non-trivial Hamming ball. The bound of $(n+o(n))^d$ is asymptotically tight. Additionally, we give a non-trivial lower bound on the size of any $d$-maximal set and show that the number of essentially different $d$-maximal sets is finite.

math.CO↗

arXiv2024

Additive Bases: Change of Domain

We consider two questions of Ruzsa on how the minimum size of an additive basis $B$ of a given set $A$ depends on the domain of $B$. To state these questions, for an abelian group $G$ and $A \subseteq D \subseteq G$ we write $\ell_D(A) \colon =\min \{ |B|: B \subseteq D, \ A \subseteq B+B \}$. Ruzsa asked how much larger can $\ell_{\mathbb{Z}}(A)$ be than $\ell_{\mathbb{Q}}(A)$ for $A\subset\mathbb{Z}$, and how much larger can $\ell_{\mathbb{N}}(A)$ be than $\ell_{\mathbb{Z}}(A)$ for $A\subset\mathbb{N}$. For the first question we show that if $\ell_{\mathbb{Q}}(A) = n$ then $\ell_{\mathbb{Z}}(A) \le 2n$, and that this is tight up to an additive error of at most $O(\sqrt{n})$. For the second question, we show that if $\ell_{\mathbb{Z}}(A) = n$ then $\ell_{\mathbb{N}}(A) \le O(n\log n)$, and this is tight up to the constant factor. We also consider these questions for higher order bases. Our proofs use some ideas that are unexpected in this context, including linear algebra and Diophantine approximation.

math.NT↗

arXiv2023

Colouring random subgraphs

We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many $k$-regular graphs $G$ for which the colouring number (i.e., degeneracy) of $G_{1/2}$ is at most $k/3 + o(k)$ with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least $k/2 - o(k)$.

math.CO↗

arXiv2023

New bounds for the same-type lemma

Given finite sets $X_1,\dotsc,X_m$ in $\mathbb{R}^d$ (with $d$ fixed), we prove that there are respective subsets $Y_1,\dotsc,Y_m$ with $|Y_i|\ge \frac{1}{\operatorname{poly}(m)}|X_i|$ such that, for $y_1\in Y_1,\dotsc,y_m\in Y_m$, the orientations of the $(d+1)$-tuples from $y_1,\dotsc,y_m$ do not depend on the actual choices of points $y_1,\dotsc,y_m$. This generalizes previously known case when all the sets $X_i$ are equal. Furthermore, we give a construction showing that polynomial dependence on $m$ is unavoidable, as well as an algorithm that approximates the best-possible constants in this result.

math.CO↗

arXiv2023

Distances between realizations of order types

Any $n$-tuple of points in the plane can be moved to any other $n$-tuple by a continuous motion with at most $\binom{n}{3}$ intermediate changes of the order type. Even for tuples with the same order type, the cubic bound is sharp: there exist pairs of $n$-tuples of the same order type requiring $c\binom{n}{3}$ intermediate changes.

math.CO↗

arXiv2023

Extremal graphs without exponentially-small bicliques

The Turán problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $Ω(n^{2-1/s})$ edges matching the upper bound of Kövári, Sós and Turán.

math.CO↗

arXiv2023

Enumeration of interval graphs and $d$-representable complexes

For each fixed $d\ge 1$, we obtain asymptotic estimates for the number of $d$-representable simplicial complexes on $n$ vertices as a function of $n$. The case $d=1$ corresponds to counting interval graphs, and we obtain new results in this well-studied case as well. Our results imply that the $d$-representable complexes comprise a vanishingly small fraction of $d$-collapsible complexes.

math.CO↗

arXiv2022

Planar convex codes are decidable

We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds on the number of vertices needed to form such realizations. Consequently we show that there is an algorithm to decide whether a convex code admits a closed or open realization in the plane.

math.CO↗

arXiv2022

Digital almost nets

Digital nets (in base $2$) are the subsets of $[0,1]^d$ that contain the expected number of points in every not-too-small dyadic box. We construct sets that contain almost the expected number of points in every such box, but which are exponentially smaller than the digital nets. We also establish a lower bound on the size of such almost nets.

math.CO↗

arXiv2022

Convex polytopes in restricted point sets in $\mathbb{R}^d$

For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) < \alpha\sqrt[d]{n}$, contains an $c$-point convex independent subset. We determine the asymptotics of $c_{d, \alpha}(n)$ as $n \to \infty$ by showing the existence of positive constants $\beta = \beta(d, \alpha)$ and $\gamma = \gamma(d)$ such that $\beta n^{\frac{d-1}{d+1}} \le c_{d, \alpha}(n) \le \gamma n^{\frac{d-1}{d+1}}$ for $\alpha\geq 2$.

math.CO↗

arXiv2021

Sharp density bounds on the finite field Kakeya problem

A Kakeya set in $\mathbb{F}_q^n$ is a set containing a line in every direction. We show that every Kakeya set in $\mathbb{F}_q^n$ has density at least $1/2^{n-1}$, matching the construction by Dvir, Kopparty, Saraf and Sudan.

math.CO↗

arXiv2021

Applications of Random Algebraic Constructions to Hardness of Approximation

In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. (*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a balanced bipartite graph with one partition class equipartitioned into k colour classes in which the common neighbourhoods of panchromatic k-sets of vertices are much larger than those of k-sets that repeat a colour. The question of their existence was raised by Karthik and Manurangsi [Combinatorica 2020]. (*) Threshold Graphs: For fixed integer k, a k-threshold graph is, roughly speaking, a balanced bipartite graph in which the common neighbourhoods of k-sets of vertices on one side are much larger than those of (k+1)-sets. The question of their existence was raised by Lin [JACM 2018]. As applications of our constructions, we show the following conditional time lower bounds on the parameterized set intersection problem where, given a collection of n sets over universe [n] and a parameter k, the goal is to find k sets with the largest intersection. (*) Assuming ETH, for any computable function F, no $n^{o(k)}$-time algorithm can approximate the parameterized set intersection problem up to factor F(k). This improves considerably on the previously best-known result under ETH due to Lin [JACM 2018], who ruled out any $n^{o(\sqrt{k})}$ time approximation algorithm for this problem. (*) Assuming SETH, for every $\varepsilon>0$ and any computable function F, no $n^{k-\varepsilon}$-time algorithm can approximate the parameterized set intersection problem up to factor F(k). No result of comparable strength was previously known under SETH, even for solving this problem exactly.

cs.CC↗

arXiv2021

Longest common subsequences between words of very unequal length

We consider the expected length of the longest common subsequence between two random words of lengths $n$ and $(1-\varepsilon)kn$ over $k$-symbol alphabet. It is well-known that this quantity is asymptotic to $γ_{k,\varepsilon} n$ for some constant $γ_{k,\varepsilon}$. We show that $γ_{k,\varepsilon}$ is of the order $1-c\varepsilon^2$ uniformly in $k$ and $\varepsilon$. In addition, for large $k$, we give evidence that $γ_{k,\varepsilon}$ approaches $1-\tfrac{1}{4}\varepsilon^2$, and prove a matching lower bound.

math.PR↗

arXiv2021

Periodic words, common subsequences and frogs

Let $W^{(n)}$ be the $n$-letter word obtained by repeating a fixed word $W$, and let $R_n$ be a random $n$-letter word over the same alphabet. We show several results about the length of the longest common subsequence (LCS) between $W^{(n)}$ and $R_n$; in particular, we show that its expectation is $γ_W n-O(\sqrt{n})$ for an efficiently-computable constant $γ_W$. This is done by relating the problem to a new interacting particle system, which we dub "frog dynamics". In this system, the particles (`frogs') hop over one another in the order given by their labels. Stripped of the labeling, the frog dynamics reduces to a variant of the PushTASEP. In the special case when all symbols of $W$ are distinct, we obtain an explicit formula for the constant $γ_W$ and a closed-form expression for the stationary distribution of the associated frog dynamics. In addition, we propose new conjectures about the asymptotic of the LCS of a pair of random words. These conjectures are informed by computer experiments using a new heuristic algorithm to compute the LCS. Through our computations, we found periodic words that are more random-like than a random word, as measured by the LCS.

math.PR↗