arXiv subjects

Oleg Pikhurko

Publications and source records attributed to Oleg Pikhurko.

At least 55 records · Page 3Linked to original sources

arXiv2023

Exact solutions to the Erdős-Rothschild problem

Let $\textbf{k} := (k_1,\ldots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\textbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\textbf{k})$ to denote the maximum of $F(G;\textbf{k})$ over all graphs $G$ on $n$ vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for $n \to \infty$: (i) A sufficient condition on $\textbf{k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results. (ii) Addressing the original question of Erdős and Rothschild, in the case $\textbf{k}=(3,\ldots,3)$ of length $7$, the unique extremal graph is the complete balanced $8$-partite graph, with colourings coming from Hadamard matrices of order $8$. (iii) In the case $\textbf{k}=(k+1,k)$, for which the sufficient condition in (i) does not hold, for $3 \leq k \leq 10$, the unique extremal graph is complete $k$-partite with one part of size less than $k$ and the other parts as equal in size as possible.

math.CO↗

arXiv2023

Large Deviation Principles for Block and Step Graphon Random Graph Models

Borgs, Chayes, Gaudio, Petti and Sen [arXiv:2007.14508] proved a large deviation principle for block model random graphs with rational block ratios. We strengthen their result by allowing any block ratios (and also establish a simpler formula for the rate function). We apply the new result to derive a large deviation principle for graph sampling from any given step graphon.

math.PR↗

arXiv2023

Stability from graph symmetrisation arguments with applications to inducibility

We present a sufficient condition for the stability property of extremal graph problems that can be solved via Zykov's symmetrisation. Our criterion is stated in terms of an analytic limit version of the problem. We show that, for example, it applies to the inducibility problem for an arbitrary complete bipartite graph $B$, which asks for the maximum number of induced copies of $B$ in an $n$-vertex graph, and to the inducibility problem for $K_{2,1,1,1}$ and $K_{3,1,1}$, the only complete partite graphs on at most five vertices for which the problem was previously open.

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arXiv2023

Forcing Generalized Quasirandom Graphs Efficiently

We study generalized quasirandom graphs whose vertex set consists of $q$ parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with at most $(10q)^q+q$ vertices; subsequently, Lovász refined the argument to show that graphs with $4(2q+3)^8$ vertices suffice. Our results imply that the structure of generalized quasirandom graphs with $q\ge 2$ parts is forced by homomorphism densities of graphs with at most $4q^2-q$ vertices, and, if vertices in distinct parts have distinct degrees, then $2q+1$ vertices suffice. The latter improves the bound of $8q-4$ due to Spencer.

math.CO↗

arXiv2023

Measurable equidecompositions for group actions with an expansion property

Given an action of a group $Γ$ on a measure space $Ω$, we provide a sufficient criterion under which two sets $A, B\subseteq Ω$ are measurably equidecomposable, i.e., $A$ can be partitioned into finitely many measurable pieces which can be rearranged using the elements of $Γ$ to form a partition of $B$. In particular, we prove that every bounded measurable subset of $R^n$, $n\ge 3$, with non-empty interior is measurably equidecomposable to a ball via isometries. The analogous result also holds for some other spaces, such as the sphere or the hyperbolic space of dimension $n\ge 2$.

math.MG↗

arXiv2023

Stability for the Erdős-Rothschild problem

Given a sequence $\mathbf{k} := (k_1,\ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. In previous work with Yilma, we constructed a finite optimisation problem whose maximum is equal to the limit of $\log_2 F(n;\mathbf{k})/{n\choose 2}$ as $n$ tends to infinity and proved a stability theorem for complete multipartite graphs $G$. In this paper we provide a sufficient condition on $\mathbf{k}$ which guarantees a general stability theorem for any graph $G$, describing the asymptotic structure of $G$ on $n$ vertices with $F(G;\mathbf{k}) = F(n;\mathbf{k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem. We apply our theorem to systematically recover existing stability results as well as all cases with $s=2$. The proof uses a novel version of symmetrisation on edge-coloured weighted multigraphs.

math.CO↗

arXiv2023

On the $(6,4)$-problem of Brown, Erdős and Sós

Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$ vertices not containing a subgraph with $k$ edges and at most $s$ vertices. In 1973, Brown, Erdős and Sós conjectured that the limit $$\lim_{n\to \infty} n^{-2} f^{(3)}(n;k+2,k)$$ exists for all $k$ and confirmed it for $k=2$. Recently, Glock showed this for $k=3$. We settle the next open case, $k=4$, by showing that $f^{(3)}(n;6,4)=\left(\frac{7}{36}+o(1)\right)n^2$ as $n\to\infty$. More generally, for all $k\in \{3,4\}$, $r\ge 3$ and $t\in [2,r-1]$, we compute the value of the limit $\lim_{n\to \infty} n^{-t}f^{(r)}(n;k(r-t)+t,k)$, which settles a problem of Shangguan and Tamo.

math.CO↗

arXiv2023

On the limit of the positive $\ell$-degree Turán problem

The minimum positive $\ell$-degree $δ^+_{\ell}(G)$ of a non-empty $k$-graph $G$ is the maximum $m$ such that every $\ell$-subset of $V(G)$ is contained in either none or at least $m$ edges of $G$; let $δ^+_{\ell}(G):=0$ if $G$ has no edges. For a family $\mathcal F$ of $k$-graphs, let $co^+ex_\ell(n,\mathcal F)$ be the maximum of $δ^+_{\ell}(G)$ over all $\mathcal F$-free $k$-graphs $G$ on $n$ vertices. We prove that the ratio $co^+ex_\ell(n,\mathcal F)/{n-\ell\choose k-\ell}$ tends to limit as $n\to\infty$, answering a question of Halfpap, Lemons and Palmer. Also, we show that the limit can be obtained as the value of a natural optimisation problem for $k$-hypergraphons; in fact, we give an alternative description of the set of possible accumulation points of almost extremal $k$-graphs.

math.CO↗

arXiv2023

Hypergraph Turán densities can have arbitrarily large algebraic degree

Grosu [On the algebraic and topological structure of the set of Turán densities. \emph{J. Combin. Theory Ser. B} \textbf{118} (2016) 137--185] asked if there exist an integer $r\ge 3$ and a finite family of $r$-graphs whose Turán density, as a real number, has (algebraic) degree greater than~$r-1$. In this note we show that, for all integers $r\ge 3$ and $d$, there exists a finite family of $r$-graphs whose Turán density has degree at least~$d$, thus answering Grosu's question in a strong form.

math.CO↗

arXiv2022

The codegree threshold of $K_4^-$

The codegree threshold $\mathrm{ex}_2(n, F)$ of a $3$-graph $F$ is the minimum $d=d(n)$ such that every $3$-graph on $n$ vertices in which every pair of vertices is contained in at least $d+1$ edges contains a copy of $F$ as a subgraph. We study $\mathrm{ex}_2(n, F)$ when $F=K_4^-$, the $3$-graph on $4$ vertices with $3$ edges. Using flag algebra techniques, we prove that if $n$ is sufficiently large then $\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4$. This settles in the affirmative a conjecture of Nagle from 1999. In addition, we obtain a stability result: for every near-extremal configuration $G$, there is a quasirandom tournament $T$ on the same vertex set such that $G$ is close in the edit distance to the $3$-graph $C(T)$ whose edges are the cyclically oriented triangles from $T$. For infinitely many values of $n$, we are further able to determine $\mathrm{ex}_2(n, K_4^-)$ exactly and to show that tournament-based constructions $C(T)$ are extremal for those values of $n$.

math.CO↗

arXiv2022

On a question of Vera T. Sós about size forcing of graphons

The $k$-sample $\mathbb{G}(k,W)$ from a graphon $W:[0,1]^2\to [0,1]$ is the random graph on $\{1,\dots,k\}$, where we sample $x_1,\dots,x_k\in [0,1]$ uniformly at random and make each pair $\{i,j\}\subseteq \{1,\dots,k\}$ an edge with probability $W(x_i,x_j)$, with all these choices being mutually independent. Let the random variable $X_k(W)$ be the number of edges in $\mathbb{G}(k,W)$. Vera T. Sós asked in 2012 whether two graphons $U,W$ are necessarily weakly isomorphic if the random variables $X_k(U)$ and $X_k(W)$ have the same distribution for every integer $k\ge 2$. This question when one of the graphons $W$ is a constant function was answered positively by Endre Csóka and independently by Jacob Fox, Tomasz Łuczak and Vera T. Sós. Here we investigate the question when $W$ is a 2-step graphon and prove that the answer is positive for a 3-dimensional family of such graphons. We also present some related results.

math.CO↗

arXiv2022

Divisibility of Spheres with Measurable Pieces

For an $r$-tuple $(γ_1,\ldots,γ_r)$ of special orthogonal $d\times d$ matrices, we say that the Euclidean $(d-1)$-dimensional sphere $S^{d-1}$ is $(γ_1,\ldots,γ_r)$-divisible if there is a subset $A\subseteq S^{d-1}$ such that its translations by the rotations $γ_1,\ldots,γ_r$ partition the sphere. Motivated by some old open questions of Mycielski and Wagon, we investigate the version of this notion where the set $A$ has to be measurable with respect to the spherical measure. Our main result shows that measurable divisibility is impossible for a "generic" (in various meanings) $r$-tuple of rotations. This is in stark contrast to the recent result of Conley, Marks and Unger which implies that, for every "generic" $r$-tuple, divisibility is possible with parts that have the property of Baire.

math.MG↗

arXiv2022

Disjoint isomorphic balanced clique subdivisions

A thoroughly studied problem in Extremal Graph Theory is to find the best possible density condition in a host graph $G$ for guaranteeing the presence of a particular subgraph $H$ in $G$. One such classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that average degree $O(k^2)$ guarantees the existence of a $K_k$-subdivision. We study two directions extending this result. On the one hand, Verstraëte conjectured that the quadratic bound $O(k^2)$ would guarantee already two vertex-disjoint isomorphic copies of a $K_k$-subdivision. On the other hand, Thomassen conjectured that for each $k \in \mathbb{N}$ there is some $d = d(k)$ such that every graph with average degree at least $d$ contains a balanced subdivision of $K_k$, that is, a copy of $K_k$ where the edges are replaced by paths of equal length. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on $d(k)$ remains open. In this paper, we show that the quadratic bound $O(k^2)$ suffices to force a balanced $K_k$-subdivision. This gives the optimal bound on $d(k)$ needed in Thomassen's conjecture and implies the existence of $O(1)$ many vertex-disjoint isomorphic $K_k$-subdivisions, confirming Verstraëte's conjecture in a strong sense.

math.CO↗

arXiv2021

Borel Combinatorics of Locally Finite Graphs

We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, local rules, Borel transversals, etc. Also, we present the construction of Andrew Marks of acyclic Borel graphs for which the greedy bound $Δ+1$ on the Borel chromatic number is best possible. In the remainder of the paper we briefly discuss various topics such as relations to LOCAL algorithms, measurable versions of Hall's marriage theorem and of Lovász Local Lemma, applications to equidecomposability, etc.

math.CO↗

arXiv2020

Asymptotic Structure for the Clique Density Theorem

The famous Erdős-Rademacher problem asks for the smallest number of $r$-cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all $r$ was determined only recently, by Reiher [Annals of Mathematics, 184 (2016) 683--707]. Here we describe the asymptotic structure of all almost extremal graphs. This task for $r=3$ was previously accomplished by Pikhurko and Razborov [Combinatorics, Probability and Computing, 26 (2017) 138--160].

math.CO↗

arXiv2020

Measurable versions of Vizing's theorem

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $Δ$ and $π$. The ``approximate'' version states that, for any Borel probability measure on the edge set and any $ε>0$, we can properly colour all but $ε$-fraction of edges with $Δ+π$ colours in a Borel way. The ``measurable'' version, which is our main result, states that if, additionally, the measure is invariant, then there is a measurable proper edge colouring of the whole edge set with at most $Δ+π$ colours.

math.CO↗

arXiv2020

Sharp bounds for decomposing graphs into edges and triangles

For a real constant $α$, let $π_3^α(G)$ be the minimum of twice the number of $K_2$'s plus $α$ times the number of $K_3$'s over all edge decompositions of $G$ into copies of $K_2$ and $K_3$, where $K_r$ denotes the complete graph on $r$ vertices. Let $π_3^α(n)$ be the maximum of $π_3^α(G)$ over all graphs $G$ with $n$ vertices. The extremal function $π_3^3(n)$ was first studied by Győri and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320]. In a recent progress on this problem, Král', Lidický, Martins and Pehova [Decomposing graphs into edges and triangles, Combin. Prob. Comput. 28 (2019) 465--472] proved via flag algebras that $π_3^3(n)\le (1/2+o(1))n^2$. We extend their result by determining the exact value of $π_3^α(n)$ and the set of extremal graphs for all $α$ and sufficiently large $n$. In particular, we show for $α=3$ that $K_n$ and the complete bipartite graph $K_{\lfloor n/2\rfloor,\lceil n/2\rceil}$ are the only possible extremal examples for large $n$.

math.CO↗

arXiv2020

The exact minimum number of triangles in graphs of given order and size

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from~$1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.

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