D. Zakharov

activity

20162026

most citedEssentially tight bounds for rainbow cycles in proper edge-colourings

3 citations · 15 across the 35 of their papers we have counts for

collaborators

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28 papers · 1 filter

math.CO2026

The Erdős distinct distances problem in $R^{3}$

Jonathan Tidor, Hung-Hsun Hans Yu, Dmitrii Zakharov

We prove that $N$ points in $R^{3}$ determine at least $N^{2/3 - o (1)}$ distinct distances.

math.CO2026

Asymptotically sharp bounds for affine subspace statistics in $F_{2}^{n}$

Ting-Wei Chao, Zixuan Xu, Dmitrii Zakharov

Given a subset $A \subseteq F_{2}^{n}$ , we can consider the distribution of the intersection size of $A$ with a uniformly random $d$ -flat $F$ . Motivated by the edge statistics…

math.CO2026

Uniform Set Systems with Uniform Witnesses

Ting-Wei Chao, Zixuan Xu, Dmitrii Zakharov

Frankl--Pach and Erdős conjectured that any $(d + 1)$ -uniform set family $F \subseteq (d + 1 [ n ])$ with VC-dimension at most $d$ has size at most $(d n - 1)$ when…

math.CO2026

An isoperimetric inequality for word overlap

Dmitrii Zakharov

Let $A$ and $B$ be sets of words of length $n$ over some finite alphabet. Suppose that no suffix of a word in $A$ coincides with a prefix of a word in $B$ . Then we show that the pr…

math.CO2025

Moderate-doubling sets in $F_{2}^{n}$ intersect subspaces

Alex Cohen, Dmitrii Zakharov

We show that any set $A$ in $F_{2}^{n}$ with $∣ A + A ∣ \leq ∣ A ∣^{2 - η}$ must intersect a subspace of dimension $O_{η} (lo g ∣ A ∣)$ in at least $∣ A ∣^{η - o (1)}$ elements.

math.CO2025

A fractal-like configuration of point-line pairs for the minimal distance problem

Alexander Logunov, Dmitrii Zakharov

We show that for every $n \in N$ there is a collection of points $p_{1}, \dots, p_{n}$ and lines $ℓ_{1}, \dots, ℓ_{n}$ in the unit square such that for any $i$ we have $p_i…

math.CO2025

Heilbronn's triangle problem in three dimensions

Dominique Maldague, Hong Wang, Dmitrii Zakharov

We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{- 2/3 - c}$ for some absolute constant $c > 0$ . This gives the first non-trivial upper bo…

math.CO2024

Generalized Arithmetic Kakeya

Cosmin Pohoata, Dmitrii Zakharov

Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a nat…

math.CO2024

Color avoidance for monotone paths

Eion Mulrenin, Cosmin Pohoata, Dmitrii Zakharov

In 2014, Moshkovitz and Shapira determined the tower height for hypergraph Ramsey numbers of tight monotone paths. We address the color-avoiding version of this problem in which on…

math.CO2024

Sharp bound for the Erdős-Straus non-averaging set problem

Huy Tuan Pham, Dmitrii Zakharov

A set of integers $A$ is non-averaging if there is no element $a$ in $A$ which can be written as an average of a subset of $A$ not containing $a$ . We show that the largest non-aver…

math.CO2024

Ruzsa's problem on Bi-Sidon sets

János Pach, Dmitrii Zakharov

A subset $S$ of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of eleme…

math.CO2024

Lower bounds for incidences

Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov

Let $p_{1}, \dots, p_{n}$ be a set of points in the unit square and let $T_{1}, \dots, T_{n}$ be a set of $δ$ -tubes such that $T_{j}$ passes through $p_{j}$ . We prove a lower bound for the numbe…

math.CO2024

An explicit economical additive basis

Vishesh Jain, Huy Tuan Pham, Mehtaab Sawhney +1

We present an explicit subset $A \subseteq N = {0, 1, \dots}$ such that $A + A = N$ and for all $ε > 0$ , \[\lim_{N\to \infty}\frac{\big|\big\{(n_1,n_2):…

math.CO2024

On skew corner-free sets

Cosmin Pohoata, Dmitrii Zakharov

We construct skew corner-free sets in $[n]^{2}$ of size $n^{5/4}$ , thereby disproving a conjecture of Kevin Pratt. We also show that any skew corner-free set in $\mathbb{F}_{q}^{n} \…

math.CO2023

A Sharp Ramsey Theorem for Ordered Hypergraph Matchings

Lisa Sauermann, Dmitrii Zakharov

We prove essentially sharp bounds for Ramsey numbers of ordered hypergraph matchings, inroduced recently by Dudek, Grytczuk, and Ruciński. Namely, for any $r \geq 2$ and $n \geq 2$ ,…

math.CO2023★ 3 cited

Essentially tight bounds for rainbow cycles in proper edge-colourings

Noga Alon, Matija Bucić, Lisa Sauermann +2

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decad…

math.CO2023★ 1 cited

A new upper bound for the Heilbronn triangle problem

Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov

For sufficiently large $n$ , we show that in every configuration of $n$ points chosen inside the unit square there exists a triangle of area less than $n^{- 8/7 - 1/2000}$ . This improv…

math.CO2023

On the Erdős--Ginzburg--Ziv Problem in large dimension

Lisa Sauermann, Dmitrii Zakharov

The Erdős--Ginzburg--Ziv Problem is a classical extremal problem in discrete geometry. Given $m$ and $n$ , the problem asks about the smallest number $s$ such that among any $s$ poi…

math.CO2022

Sharp bounds for rainbow matchings in hypergraphs

Cosmin Pohoata, Lisa Sauermann, Dmitrii Zakharov

Suppose we are given matchings $M_{1}, ...., M_{N}$ of size $t$ in some $r$ -uniform hypergraph, and let us think of each matching having a different color. How large does $N$ need to be…

math.CO2022★ 1 cited

Cutting corners

Andrey Kupavskii, Arsenii Sagdeev, Dmitrii Zakharov

We say that a subset $M$ of $R^{n}$ is exponentially Ramsey if there are $ε > 0$ and $n_{0}$ such that $χ (R^{n}, M) \geq (1 + ε)^{n}$ for any $n > n_{0}$ , where $χ (R^{n}, M)$ stan…

math.CO2022★ 1 cited

Upper bounds for Heilbronn's triangle problem in higher dimensions

Dmitrii Zakharov

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for…

math.CO2022★ 2 cited

Convex polytopes from fewer points

Cosmin Pohoata, Dmitrii Zakharov

Let $E S_{d} (n)$ be the smallest integer such that any set of $E S_{d} (n)$ points in $R^{d}$ in general position contains $n$ points in convex position. In 1960, Erdős and S…

math.CO2022★ 1 cited

Spread approximations for forbidden intersections problems

Andrey Kupavskii, Dmitriy Zakharov

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 `spr…

math.CO2021★ 2 cited

On the number of high-dimensional partitions

Cosmin Pohoata, Dmitriy Zakharov

Let $P_{d} (n)$ denote the number of $n \times \dots \times n$ $d$ -dimensional partitions with entries from ${0, 1, \dots, n}$ . Building upon the works of Balogh-Treglow…

The Erdős distinct distances problem in R3

Asymptotically sharp bounds for affine subspace statistics in F2n​

Uniform Set Systems with Uniform Witnesses

An isoperimetric inequality for word overlap

Moderate-doubling sets in F2n​ intersect subspaces

A fractal-like configuration of point-line pairs for the minimal distance problem

Heilbronn's triangle problem in three dimensions

Generalized Arithmetic Kakeya

Color avoidance for monotone paths

Sharp bound for the Erdős-Straus non-averaging set problem

Ruzsa's problem on Bi-Sidon sets

Lower bounds for incidences

An explicit economical additive basis

On skew corner-free sets

A Sharp Ramsey Theorem for Ordered Hypergraph Matchings

Essentially tight bounds for rainbow cycles in proper edge-colourings

A new upper bound for the Heilbronn triangle problem

On the Erdős--Ginzburg--Ziv Problem in large dimension

Sharp bounds for rainbow matchings in hypergraphs

Cutting corners

Upper bounds for Heilbronn's triangle problem in higher dimensions

Convex polytopes from fewer points

Spread approximations for forbidden intersections problems

On the number of high-dimensional partitions

The Erdős distinct distances problem in $R^{3}$

Asymptotically sharp bounds for affine subspace statistics in $F_{2}^{n}$

Moderate-doubling sets in $F_{2}^{n}$ intersect subspaces