arXiv subjects

Oleg Pikhurko

Publications and source records attributed to Oleg Pikhurko.

At least 37 records · Page 2Linked to original sources

arXiv2025

Circle Squaring with Pieces of Small Boundary and Low Borel Complexity

Tarski's Circle Squaring Problem from 1925 asks whether it is possible to partition a disk in the plane into finitely many pieces and reassemble them via isometries to yield a partition of a square of the same area. It was finally resolved by Laczkovich in 1990 in the affirmative. Recently, several new proofs have emerged which achieve circle squaring with better structured pieces: namely, pieces which are Lebesgue measurable and have the property of Baire (Grabowski-Máthé-Pikhurko) or even are Borel (Marks-Unger). In this paper, we show that circle squaring is possible with Borel pieces of positive Lebesgue measure whose boundaries have upper Minkowski dimension less than 2 (in particular, each piece is Jordan measurable). We also improve the Borel complexity of the pieces: namely, we show that each piece can be taken to be a Boolean combination of $F_σ$ sets. This is a consequence of our more general result that applies to any two bounded subsets of $R^k$, $k\ge 1$, of equal positive measure whose boundaries have upper Minkowski dimension smaller than $k$.

math.MG↗

arXiv2025

On the quadratic 8-edge case of the Brown-Erdős-Sós problem

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ edges on at most $s$ vertices. Brown, Erdős and Sós conjectured in 1973 that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. Recently, Delcourt and Postle settled the conjecture and their approach was generalised by Shangguan to every uniformity $r\ge 4$: the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ exists for all $r\ge 3$ and $k\ge 2$. The value of the limit is currently known for $k\in \{2,3,4,5,6,7\}$ due to various results authored by Glock, Joos, Kim, Kühn, Lichev, Pikhurko, Rödl and Sun. In this paper we consider the case $k=8$, determining the value of the limit for each $r\ge 4$ and presenting a lower bound for $k=3$ that we conjecture to be sharp.

math.CO↗

arXiv2025

Some exact values of the inducibility and statistics constants for hypercubes

We consider two types of problems: maximising, over subsets $S\subseteq \{0,1\}^n$, the density of $d$-subcubes $C$ in the $n$-hypercube graph that span a subgraph such that $S\cap C$ is i) isomorphic to the given configuration $H\subseteq\{0,1\}^d$ (the inducibility problem), or ii) has the given size $s$ (the statistics problem). Using flag algebras, we determine the limit of this density as $n\to\infty$ for 5 new configurations $H\subseteq\{0,1\}^3$ and for 3 new pairs $(d,s)$, namely for $(3,2)$, $(4,2)$ and $(4,4)$. Interestingly, the lower bounds in the last three cases come from blowups of small Hamming codes.

math.CO↗

arXiv2025

New upper bound for lattice covering by spheres

We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^β n \big)$ as $n\to\infty$, where \begin{align*} β:= \frac{1}{2} \log_2 \left(\frac{8 π\mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O\big(n \ln^α n \big)$, where $α:= \frac{1}{2} \log_{2}(2π\mathrm{e})=2.0471...\,.$

math.MG↗

arXiv2025

The Turán density of the tight 5-cycle minus one edge

Let the tight $\ell$-cycle minus one edge $C_\ell^{3-}$ be the $3$-graph on $\{1,\dots,\ell\}$ consisting of $\ell-1$ consecutive triples in the cyclic order. We show that, for every $\ell\ge 5$ not divisible by $3$, the Turán density of $C_{\ell}^{3-}$ is $1/4$ and also prove some finer structure results. This proves a conjecture of Mubayi--Sudakov--Pikhurko from 2011 and extends the results of Balogh--Luo [Combinatorica 44 (2024) 949--976] who established analogous claims for all sufficiently large $\ell$. Results similar to ours were independently obtained by Lidický--Mattes--Pfender [arXiv:2409.14257].

math.CO↗

arXiv2025

The Turán density of short tight cycles

The $3$-uniform tight $\ell$-cycle $C_\ell^{3}$ is the $3$-graph on $\{1,\dots,\ell\}$ consisting of all $\ell$ consecutive triples in the cyclic order. Let $\mathcal{C}$ be either the pair $\{C_{4}^{3}, C_{5}^{3}\}$ or the single tight $\ell$-cycle $C_{\ell}^{3}$ for some $\ell\ge 7$ not divisible by $3$. We show that the Turán density of $\mathcal{C}$, that is, the asymptotically maximal edge density of a large $\mathcal{C}$-free $3$-graph, is equal to $2\sqrt{3} - 3$. We also establish the corresponding Erdős-Simonovits-type stability result, informally stating that all almost maximum $\mathcal{C}$-free graphs are close in the edit distance to a 2-part recursive construction. This extends the earlier analogous results of Kamčev-Letzter-Pokrovskiy ["The Turán density of tight cycles in three-uniform hypergraphs", Int. Math. Res. Not. 6 (2024), 4804-4841] that apply for sufficiently large $\ell$ only. Additionally, we prove a finer structural result that allows us to determine the maximum number of edges in a $\{C_{4}^{3}, C_{5}^{3}\}$-free $3$-graph with a given number of vertices up to an additive $O(1)$ error term.

math.CO↗

arXiv2025

Large deviation principles for graphon sampling

We investigate possible large deviation principles (LDPs) for the $n$-vertex sampling from a given graphon with various speeds $s(n)$ and resolve all the cases except when the speed $s(n)$ is of order $n^2$. For quadratic speed $s=(c+o(1))n^2$, we establish an LDP for an arbitrary $k$-step graphon, which extends a result of Chatterjee and Varadhan [Europ. J. Combin., 32 (2011) 1000-1017] who did this for $k=1$ (that is, for the homogeneous binomial random graphs). This is done by reducing the problem to the LDP for stochastic $k$-block models established recently by Borgs, Chayes, Gaudio, Petti and Sen ["A large deviation principle for block models", arxiv:2007.14508, 2020]. Also, we improve some results by Borgs et al.

math.PR↗

arXiv2025

Phase transition of degenerate Turán problems in $p$-norms

For a positive real number $p$, the $p$-norm $\left\lVert G \right\rVert_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices. Füredi and Kündgen \cite{FK06} show that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $\mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions, assuming a mild assumption on the growth rate of $\mathrm{ex}(n,F)$. The main contribution of our paper is extending this result to hypergraph. Moreover, in the case of graph, our proof differs from that in \cite{FK06}, offering the advantage of producing the correct constant factor when $p > p_{F}$. When $p = p_F$, Füredi and Kündgen proved a general upper bound on $\mathrm{ex}_{p}(n,F)$, tight up to a $\log n$ factor, and conjectured that this factor is unnecessary. We confirm this conjecture for several well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles.

math.CO↗

arXiv2025

On the $(k+2,k)$-problem of Brown, Erdős and Sós for $k=5,6,7$

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no subgraph with $k$ edges and at most $s$ vertices. Brown, Erdős and Sós [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press 1973] conjectured that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. The value of the limit was previously determined for $k=2$ in the original paper of Brown, Erdős and Sós, for $k=3$ by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for $k=4$ by Glock, Joos, Kim, Kühn, Lichev and Pikhurko [Proc. Amer. Math. Soc., Series B, 11 (2024) 173-186] while Delcourt and Postle [Proc. Amer. Math. Soc., 152 (2024), 1881-1891] proved the conjecture (without determining the limiting value). In this paper, we determine the value of the limit in the Brown-Erdős-Sós Problem for $k\in \{5,6,7\}$. More generally, we obtain the value of $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all $r\geq 3$ and $k\in \{5,6,7\}$. In addition, by combining these new values with recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new asymptotic values for several generalised Ramsey numbers.

math.CO↗

arXiv2025

Constructions of Turán systems that are tight up to a multiplicative constant

For positive integers $n\ge s> r$, the Turán function $T(n,s,r)$ is the smallest size of an r-graph with n vertices such that every set of s vertices contains at least one edge. Also, define the Turán density $t(s,r)$ as the limit of $T(n,s,r)/ {n\choose r}$ as $n\to\infty$. The question of estimating these parameters received a lot of attention after it was first raised by Turán in 1941. A trivial lower bound is $t(s,r)\ge 1/{s\choose s-r}$. In the early 1990s, de Caen conjectured that $r\cdot t(r+1,r)\to\infty$ as $r\to\infty$ and offered 500 Canadian dollars for resolving this question. We disprove this conjecture by showing more strongly that for every integer $R\ge1$ there is $μ_R$ (in fact, $μ_R$ can be taken to grow as $(1+o(1))\, R\ln R$) such that $t(r+R,r)\le (μ_R+o(1))/ {r+R\choose R}$ as $r\to\infty$, that is, the trivial lower bound is tight for every $R$ up to a multiplicative constant $μ_R$.

math.CO↗

arXiv2025

A note on the minimum size of Turán systems

For positive integers $n \ge s > r$, a \emph{Turán $(n,s,r)$-system} is an $n$-vertex $r$-graph in which every set of $s$ vertices contains at least one edge. Let $T(n,s,r)$ denote the the minimum size of a Turán $(n,s,r)$-system. Upper bounds on $T(n,s,r)$ were established by Sidorenko~\cite{Sid97} for the case $s-r = Ω(r/\ln r)$ (based on a construction of Frankl--Rödl~\cite{FR85}) and by a number of authors in the case $s-r = O(1)$. In this note, we establish upper bounds in the remaining range $O(1)<s-r = O(r/\ln r)$.

math.CO↗

arXiv2024

A note on extremal constructions for the Erdős--Rademacher problem

For given positive integers $r\ge 3$, $n$ and $e\le \binom{n}{2}$, the famous Erd\H os--Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lovász and Simonovits from the 1970s states that, for every $r\ge 3$, if $n$ is sufficiently large then, for every $e\le \binom{n}{2}$, at least one extremal graph can be obtained from a complete partite graph by adding a triangle-free graph into one part. In this note, we explicitly write the minimum number of $r$-cliques predicted by the above conjecture. Also, we describe what we believe to be the set of extremal graphs for any $r\ge 4$ and all large~$n$, amending the previous conjecture of Pikhurko and Razborov.

math.CO↗

arXiv2024

Finite Hypergraph Families with Rich Extremal Turán Constructions via Mixing Patterns

We prove that, for any finite set of minimal $r$-graph patterns, there is a finite family $\mathcal F$ of forbidden $r$-graphs such that the extremal Turán constructions for $\mathcal F$ are precisely the maximum $r$-graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author \cite{PI14}, where the above statement was established for a single pattern. We present two applications of this result. First, we construct a finite family $\mathcal F$ of $3$-graphs such that there are exponentially many maximum $\mathcal F$-free $3$-graphs of each large order $n$ and, moreover, the corresponding Turán problem is not finitely stable. Second, we show that there exists a finite family $\mathcal{F}$ of $3$-graphs whose feasible region function attains its maximum on a Cantor-type set of positive Hausdorff dimension.

math.CO↗

arXiv2024

New lower bound on ball packing density in high-dimensional hyperbolic spaces

We present a new lower bound on the Bowen-Radin maximal density of radius-R ball packings in the m-dimensional hyperbolic space, improving on the basic covering bound by factor Ω(m(R+\ln m)) as m tends to infinity. This is done by applying the recent theorem of Campos, Jenssen, Michelen and Sahasrabudhe on independent sets in graphs with sparse neighbourhoods.

math.CO↗

arXiv2024

Nondegenerate Turán problems under $(t,p)$-norms

Given integers $r > t \ge 1$ and a real number $p > 0$, the $(t,p)$-norm $\left\lVert \mathcal{H} \right\rVert_{t,p}$ of an $r$-graph $\mathcal{H}$ is the sum of the $p$-th power of the degrees $d_{\mathcal{H}}(T)$ over all $t$-subsets $T \subset V(\mathcal{H})$. We conduct a systematic study of the Turán-type problem of determining $\mathrm{ex}_{t,p}(n,\mathcal{F})$, which is the maximum of $\left\lVert \mathcal{H} \right\rVert_{t,p}$ over all $n$-vertex $\mathcal{F}$-free $r$-graphs $\mathcal{H}$. We establish several basic properties for the $(t,p)$-norm of $r$-graphs, enabling us to derive general theorems from the recently established framework in~\cite{CL24} that are useful for determining $\mathrm{ex}_{t,p}(n,\mathcal{F})$ and proving the corresponding stability. We determine the asymptotic value of $\mathrm{ex}_{t,p}(n,H_{F}^{r})$ for all feasible combinations of $(r,t,p)$ and for every graph $F$ with chromatic number greater than $r$, where $H_{F}^{r}$ represents the expansion of $F$. In the case where $F$ is edge-critical and $p \ge 1$, we establish strong stability and determine the exact value of $\mathrm{ex}_{t,p}(n,H_{F}^{r})$ for all sufficiently large $n$. These results extend the seminal theorems of Erdős--Stone--Simonovits, Andrásfai--Erdős--Sós, Erdős--Simonovits, and a classical theorem of Mubayi. For the $3$-uniform generalized triangle $F_5$, we determine the exact value of $\mathrm{ex}_{2,p}(n,F_5)$ for all $p \ge 1$ and its asymptotic value for all $p \in [1/2, 1]\cup \{k^{-1} \colon k \in 6\mathbb{N}^{+}+\{0,2\}\}$. This extends old theorems of Bollobás, Frankl--Füredi, and a recent result of Balogh--Clemen--Lidický. Our proofs utilize results on the graph inducibility problem, Steiner triple systems, and the feasible region problem introduced by Liu--Mubayi.

math.CO↗

arXiv2024

Local version of Vizing's theorem for multi-graphs

Extending a result of Christiansen, we prove that every mutli-graph $G=(V,E)$ admits a proper edge colouring $ϕ:E\to \{1,2,\dots\}$ which is local, that is, $ϕ(e)\le \max\{d(x)+π(x),d(y)+π(y)\}$ for every edge $e$ with end-points $x,y\in V$, where $d(z)$ (resp.\ $π(z)$) denotes the degree of a vertex $z$ (resp.\ the maximum edge multiplicity at $z$). This is derived from a local version of the Fan Equation.

math.CO↗

arXiv2024

Borel version of the Local Lemma

We prove a Borel version of the local lemma, i.e. we show that, under suitable assumptions, if the set of variables in the local lemma has a structure of a Borel space, then there exists a satisfying assignment which is a Borel function. The main tool which we develop for the proof, which is of independent interest, is a parallel version of the Moser-Tardos algorithm which uses the same random bits to resample clauses that are far enough in the dependency graph.

math.CO↗

arXiv2023

New lower bounds on kissing numbers and spherical codes in high dimensions

Let the kissing number $K(d)$ be the maximum number of non-overlapping unit balls in $\mathbb R^d$ that can touch a given unit ball. Determining or estimating the number $K(d)$ has a long history, with the value of $K(3)$ being the subject of a famous discussion between Gregory and Newton in 1694. We prove that, as the dimension $d$ goes to infinity, $$ K(d)\ge (1+o(1)){\frac{\sqrt{3π}}{4\sqrt2}}\,\log\frac{3}{2}\cdot d^{3/2}\cdot \Big(\frac{2}{\sqrt{3}}\Big)^{d}, $$ thus improving the previously best known bound of Jenssen, Joos and Perkins by a factor of $\log(3/2)/\log(9/8)+o(1)=3.442...$. Our proof is based on the novel approach from Jenssen, Joos and Perkins that uses the hard core sphere model of an appropriate fugacity. Similar constant-factor improvements in lower bounds are also obtained for general spherical codes, as well as for the expected density of random sphere packings in the Euclidean space $\mathbb R^d$.

math.CO↗