arXiv subjects

Oleg Pikhurko

Publications and source records attributed to Oleg Pikhurko.

At least 19 recordsLinked to original sources

arXiv2026

An improved bound on the minimum size of Tur\'an $(r+1,r)$-systems

For positive integers $n\ge s>r$, let $T(n,s,r)$ denote the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices such that every $s$-set of vertices contains at least one edge. A simple averaging argument shows that the ratio $T(n,s,r)/\binom nr$ is non-decreasing in $n$ and we denote its limit as $n\to\infty$ by $t(s,r)$. The case $s=r+1$ has a rich history, with the previously best known asymptotic bounds for $r\to\infty$ being $1\le r\cdot t(r+1,r)\le 4.91...$ . In this paper, we present a simple probabilistic construction which shows that $(r+2)\cdot t(r+1,r)\le 4$ for every $r\ge1$. We also derandomise it and discuss applications to covering codes.

math.CO↗

arXiv2026

Intervals of uniform Tur\'an densities

We prove that the set $\Pi_{\therefore,\infty}$ of uniform Tur\'an densities of possibly infinite families of $3$-graphs contains a terminal interval: there exists $\delta>0$ such that $[1-\delta,1]\subseteq\Pi_{\therefore,\infty}$. Consequently, $\Pi_{\therefore,\infty}$ has positive Lebesgue measure and Hausdorff dimension $1$.

math.CO↗

arXiv2026

Strong non-principality of positive codegree Tur\'an density

The \emph{minimum positive codegree} $\delta^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The \emph{positive codegree Tur\'an density} of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $\delta^+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0<\gamma^+(F_1, F_2) < \min\{\gamma^+(F_1), \gamma^+(F_2)\}. $$

math.CO↗

arXiv2026

A note on the Ratio and Inertia Bounds for the $k$-Independence Number

The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875--2885] proposed extensions of the classical ratio (for regular graphs) and inertia bounds to the independence number of $G^k$ for $k\ge 2$. Continuing a line of work comparing these two parameters with other known bounds, we show that the $\vartheta$-function of L. Lov\'asz and the weighted inertia bound of A. R. Calderbank and P. Frankl, when applied directly to $G^k$, perform at least as well as the ratio and inertia bounds of Abiad-Coutinho-Fiol, respectively. In particular, $\vartheta(G^k)$ provides a polynomial-time computable upper bound on the independence number of $G^k$ that is at least as strong as the ratio bound when the latter applies (i.e.,\ when the graph $G$ is regular).

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arXiv2026

The inducibility of 6-vertex graphs

The inducibility constant $\lambda_{F}$ of a graph $F$ is the asymptotically maximum induced density of $F$ in a growing sequence of graphs. This paper systematically investigates the case when $F$ has 6 vertices (and there are 78 cases to consider up to isomorphism and complementation). We show that flag algebras can compute the sharp upper bound on $\lambda_F$ in 36 cases of which, as far as the authors know, 30 are new results. In each of the solved cases, we also prove results about the structure of large (almost) extremal graphs. In particular, we establish perfect stability in all 32 cases when the extremal construction has no quasirandom parts. We also present conjectures about the value of $\lambda_{F}$ for 12 further cases (where the upper and lower bounds are very close to each other).

math.CO↗

arXiv2026

Intervals of hypergraph Tur\'an densities

We prove that, for every integer $r\ge 3$, the set $\Pi^{(r)}_\infty$ of Tur\'an densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-\delta_r,1]$ for some $\delta_{r}>0$. This answers a question of Frankl, Peng, R\"odl and Talbot from 2007. This also shows that the Hausdorff dimension of $\Pi^{(r)}_\infty$ has the maximum possible value 1, thus resolving a question of Grosu from 2016, whereas previously it was not even known whether it is non-zero. We also derive that the set of uniform Tur\'an densities of finite families of $3$-graphs is dense in a non-degenerate interval.

math.CO↗

arXiv2026

On the spectral gap conjecture for pairs in SU(2)

For $n \ge 2$, Gamburd, Jakobson, and Sarnak [J. Eur. Math. Soc. 1, 51-85 (1999)] conjectured that almost every $n$-tuple in $\mathrm{SU}(2)$ has a spectral gap. Toward this conjecture, Fisher [Int. Math. Res. Not. (2006)] established a zero-one law for $n \ge 3$, but obtained only a partial result for $n=2$. In this paper, we prove that the zero-one law also holds for $n=2$. We also remark that a Baire categorical analogue of this result holds.

math.GR↗

arXiv2026

On problems of Erd\H{o}s and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs

Using flag algebras, we prove that the minimum density of $8$-cliques in a large graph without an independent set of size $3$ is $491411/268435456+o(1)$, thus resolving a new case of an old problem of Erd\H{o}s [Magyar Tud. Akad. Mat. Kutat\'o Int. K\"ozl. 7 (1962) 459-464]. Also, we establish some other results of this type; for example, we show that the minimum $s$-clique density in a large graph with no independent set of size 3 nor an induced 5-cycle is $2^{1-s}+o(1)$ when $s=4,5,6$. For each of these results, we also describe the structure of all extremal and almost extremal graphs of large order $n$. These results are applied to give an asymptotic solution to a number of cases of the problem of Baumann and Briggs [Electronic J Comb 32 (2025) P1.22] which asks for the minimum number of $s$-cliques in an $n$-vertex graph in which every $k$-set spans a $t$-clique.

math.CO↗

arXiv2026

Convergence of spectra of digraph limits

The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon.

math.CO↗

arXiv2026

Rational codegree Turán density of hypergraphs

Let $H$ be a $k$-graph (i.e. a $k$-uniform hypergraph). Its minimum codegree $δ_{k-1}(H)$ is the largest integer $t$ such that every $(k-1)$-subset of $V(H)$ is contained in at least $t$ edges of~$H$. The \emph{codegree Turán density} $γ(\mathcal{F})$ of a family $\mathcal{F}$ of $k$-graphs is the infimum of $γ> 0$ such that every $k$-graph $H$ on $n\to\infty$ vertices with $δ_{k-1}(H) \ge (γ+o(1))\, n$ contains some member of $\mathcal{F}$ as a subgraph. We prove that, for every integer $k\ge3$ and every rational number $α\in [0,1)$, there exists a finite family of $k$-graphs $\mathcal{F}$ such that $γ(\mathcal{F})=α$. Also, for every $k \ge 3$, we establish a strong version of non-principality, namely that there are two $k$-graphs $F_1$ and $F_2$ such that the codegree Turán density of $\{F_1,F_2\}$ is strictly smaller than that of each $F_i$. This answers a question of Mubayi and Zhao [J Comb Theory (A) 114 (2007) 1118--1132].

math.CO↗

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

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

Semi-inducibility of 4-vertex graphs

For a graph $H$ whose edges are coloured blue or red, the $H$-semi-inducibility problem asks for the maximum, over all graphs $G$ of given order $n$, of the number of injections from the vertex set of $H$ into the vertex set of $G$ that send red (resp. blue) edges of $H$ to edges (resp. non-edges) of $G$. We consider all possible 4-vertex non-complete graphs $H$ and essentially resolve all remaining cases except when $H$ is the 3-edge path coloured blue-blue-red in this order (or is equivalent to this case). Some of our proofs are computer-generated, using the flag algebra method of Razborov.

math.CO↗

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

Some exact inducibility-type results for graphs via flag algebras

The $(\kappa,\ell)$-edge-inducibility problem asks for the maximum number of $\kappa$-subsets inducing exactly $\ell$ edges that a graph of given order $n$ can have. Using flag algebras and stability approach, we resolve this problem for all sufficiently large $n$ (including a description of all extremal and almost extremal graphs) in eleven new non-trivial cases when $\kappa\le 7$. We also compute the $F$-inducibility constant (the asymptotically maximum density of induced copies of $F$ in a graph of given order $n$) and obtain some corresponding structure results for three new graphs $F$ with $5$ vertices: the 3-edge star plus an isolated vertex, the 4-cycle plus an isolated vertex, and the 4-cycle with a pendant edge.

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