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Dmitriy Zakharov

Publications and source records attributed to Dmitriy Zakharov.

11 recordsLinked to original sources
arXiv2024

Spread approximations for forbidden intersections problems

We develop a new approach to approximate families of sets, complementing the existing `$Δ$-system method' and `junta approximations method'. The approach, which we refer to as `spread approximations method', is based on the notion of $r$-spread families and builds on the recent breakthrough result of Alweiss, Lovett, Wu and Zhang for the Erd\H os--Rado `Sunflower Conjecture'. Our approach can work in a variety of sparse settings. To demonstrate the versatility and strength of the approach, we present several of its applications to forbidden intersection problems, including bounds on the size of regular intersecting families, the resolution of the Erd\H os--Sós problem for sets in a new range and, most notably, the resolution of the $t$-intersection and Erd\H os--Sós problems for permutations in a new range. Specifically, we show that any collection of permutations of an $n$-element set with no two permutations intersecting in at most (exactly) $t-1$ elements has size at most $(n-t)!$, provided $t\le n^{1-ε}$ ($t \le n^{\frac{1}{3}-ε}$) for an arbitrary $ε>0$ and $n>n_0(ε)$. Previous results for these problems only dealt with the case of fixed $t$. The proof follows the structure vs. randomness philosophy, which proved to be very efficient in proving results throughout mathematics and computer science.

math.CO↗
arXiv2024

On the number of high-dimensional partitions

Let $P_{d}(n)$ denote the number of $n \times \ldots \times n$ $d$-dimensional partitions with entries from $\left\{0,1,\ldots,n\right\}$. Building upon the works of Balogh-Treglown-Wagner and Noel-Scott-Sudakov, we show that when $d \to \infty$, $$P_{d}(n) = 2^{(1+o_{d}(1)) \sqrt{\frac{6}{(d+1)π}} \cdot n^{d}}$$ holds for all $n \geq 1$. This makes progress towards a conjecture of Moshkovitz-Shapira [{\it{Adv. in Math.}} 262 (2014), 1107--1129]. Via the main result of Moshkovitz and Shapira, our estimate also determines asymptotically a Ramsey theoretic parameter related to Erdős-Szekeres-type functions, thus solving a problem of Fox, Pach, Sudakov, and Suk [{\it{Proc. Lond. Math. Soc.}} 105 (2012), 953--982]. Our main result is a new supersaturation theorem for antichains in $[n]^{d}$, which may be of independent interest.

math.CO↗
arXiv2021

Random multilinear maps and the Erdős box problem

By using random multilinear maps, we provide new lower bounds for the Erdős box problem, the problem of estimating the extremal number of the complete $d$-partite $d$-uniform hypergraph with two vertices in each part, thereby improving on work of Gunderson, Rödl and Sidorenko.

math.CO↗
arXiv2021

Zero subsums in vector spaces over finite fields

The Olson constant $\mathcal{O}L(\mathbb{F}_{p}^{d})$ represents the minimum positive integer $t$ with the property that every subset $A\subset \mathbb{F}_{p}^{d}$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $\mathcal{O}L(\mathbb{F}_{p}^{d})$ is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case $d=1$. In this paper, we prove that for any fixed $d \geq 2$ and $ε> 0$, the Olson constant of $\mathbb{F}_{p}^{d}$ satisfies the inequality $$\mathcal{O}L(\mathbb{F}_{p}^{d}) \leq (d-1+ε)p$$ for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.

math.CO↗
arXiv2021

Norm hypergraphs

We introduce a high uniformity generalization of the so-called (projective) norm graphs of Alon, Kollár, Rónyai, and Szabó, and use it to show that $$\operatorname{ex}_{d}(n,K_{s_{1},\ldots,s_{d}}^{(d)}) = Θ\left(n^{d - \frac{1}{s_{1}\ldots s_{d-1}}}\right)$$ holds for all integers $s_{1},\ldots,s_{d} \geq 2$ such that $s_{d} \geq \left((d-1)(s_{1}\ldots s_{d-1}-1)\right)!+1$. This improves upon a recent result of Ma, Yuan and Zhang, and thus settles (many) new cases of a conjecture of Mubayi.

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arXiv2020

The extremal number of surfaces

In 1973, Brown, Erdős and Sós proved that if $\mathcal{H}$ is a 3-uniform hypergraph on $n$ vertices which contains no triangulation of the sphere, then $\mathcal{H}$ has at most $O(n^{5/2})$ edges, and this bound is the best possible up to a constant factor. Resolving a conjecture of Linial, also reiterated by Keevash, Long, Narayanan, and Scott, we show that the same result holds for triangulations of the torus. Furthermore, we extend our result to every closed orientable surface $\mathcal{S}$.

math.CO↗
arXiv2020

Turán-type results for intersection graphs of boxes

In this short note, we prove the following analog of the Kővári-Sós-Turán theorem for intersection graphs of boxes. If $G$ is the intersection graph of $n$ axis-parallel boxes in $\mathbb{R}^{d}$ such that $G$ contains no copy of $K_{t,t}$, then $G$ has at most $ctn(\log n)^{2d+3}$ edges, where $c=c(d)>0$ only depends on $d$. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit et al. of $K_{2,2}$-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon et al. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

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arXiv2020

The right acute angles problem?

The Danzer--Grünbaum acute angles problem asks for the largest size of a set of points in ${\mathbb R}^d$ that determines only acute angles. Recently, the problem was essentially solved thanks to the results of the second author and of Gerencsér and Harangi: now, the lower and the upper bounds are $2^{d-1}+1$ and $2^d-1$, respectively. The lower-bound construction is surprisingly simple. In this note, we suggest the following variant of the problem, which is one way to "save" the problem. Put $F(α) = \lim_{d\to \infty} f(d,α)^{1/d}$, where $f(d,α)$ is the largest set of points in ${\mathbb R}^d$ with no angle greater than $α$. Then the question is to find $c:= \lim_{α\to π/2^-}F(α).$ Although one may expect that $c=2$ in view of the result of Gerencsér and Harangi, the best lower bound we could get is $c\ge \sqrt 2$. We also solve a related problem of Erdos and Füredi on the "stability" of the acute angles problem and refute another conjecture stated in the same paper.

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arXiv2019

Chromatic numbers of Kneser-type graphs

Let $G(n, r, s)$ be a graph whose vertices are all $r$-element subsets of an $n$-element set, in which two vertices are adjacent if they intersect in exactly $s$ elements. In this paper we study chromatic numbers of $G(n, r, s)$ with $r, s$ being fixed constants and $n$ tending to infinity. Using a recent result of Keevash on existence of designs we deduce an inequality $χ(G(n, r, s)) \le (1+o(1))n^{r-s} \frac{(r-s-1)!}{(2r-2s-1)!}$ for $r > s$ with $r, s$ fixed constants. This inequality gives sharp upper bounds for $r \le 2s+1$. Also we develop an elementary approach to this problem and prove that $χ(G(n, 4, 2)) \sim \frac{n^2}{6}$ without use of Keevash's results. Some bounds on the list chromatic number of $G(n, r, s)$ are also obtained.

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arXiv2017

Regular bipartite graphs and intersecting families

In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H os--Ko--Rado theorem, the Hilton--Milner theorem, a theorem due to Frankl concerning the size of intersecting families with bounded maximal degree, and versions of results on the sum of sizes of non-empty cross-intersecting families due to Frankl and Tokushige. Several new stronger results are also obtained. Our approach is based on the use of regular bipartite graphs. These graphs are quite often used in Extremal Set Theory problems, however, the approach we develop proves to be particularly fruitful.

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