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Vojtěch Rödl

Publications and source records attributed to Vojtěch Rödl.

At least 37 records · Page 2Linked to original sources
arXiv2021

A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

We show that for every $r \ge 2$ there exists $ε_r > 0$ such that any $r$-uniform hypergraph with $m$ edges and maximum vertex degree $o(\sqrt{m})$ contains a set of at most $(\frac{1}{2} - ε_r)m$ edges the removal of which breaks the hypergraph into connected components with at most $m/2$ edges. We use this to give an algorithm running in time $d^{(1 - ε_r)m}$ that decides satisfiability of $m$-variable $(d, k)$-CSPs in which every variable appears in at most $r$ constraints, where $ε_r$ depends only on $r$ and $k\in o(\sqrt{m})$. Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable $(2, k)$-CSPs with variable frequency $r$ can be refuted in tree-like resolution in size $2^{(1 - ε_r)m}$. Furthermore for Tseitin formulas on graphs with degree at most $k$ (which are $(2, k)$-CSPs) we give a deterministic algorithm finding such a refutation.

cs.LO↗
arXiv2021

Independent sets in subgraphs of a shift graph

Erdős, Hajnal and Szemerédi proved that any subset $G$ of vertices of a shift graph $\text{Sh}_{n}^{k}$ has the property that the independence number of the subgraph induced by $G$ satisfies $α(\text{Sh}_{n}^{k}[G])\geq \left(\frac{1}{2}-\varepsilon\right)|G|$, where $\varepsilon\to 0$ as $k\to \infty$. In this note we prove that for $k=2$ and $n \to \infty$ there are graphs $G\subseteq \binom{[n]}{2}$ with $α(\text{Sh}_{n}^{2}[G])\leq \left(\frac{1}{4}+o(1)\right)|G|$, and $\frac{1}{4}$ is best possible. We also consider a related problem for infinite shift graphs.

math.CO↗
arXiv2021

Every Steiner triple system contains an almost spanning d-ary hypertree

In this paper we make a partial progress on the following conjecture: for every $μ>0$ and large enough $n$, every Steiner triple system $S$ on at least $(1+μ)n$ vertices contains every hypertree $T$ on $n$ vertices. We prove that the conjecture holds if $T$ is a perfect $d$-ary hypertree.

math.CO↗
arXiv2021

Colourful matchings

Suppose a committee consisting of three members has to match $n$ candidates to $n$ different positions. Each member of the committee proposes a matching, however the proposed matchings totally disagree, i.e., every candidate is matched to three different positions according to three committee members. All three committee members are very competitive and want to push through as many of their suggestions as possible. Can a committee always find a compromise -- a matching of candidates to positions such that for every committee member a third of all candidates are assigned according to that committee member suggestion? We will consider an asymptotic version of this question and several other variants of similar problem. As an application we will consider an embedding problem -- in particular which configurations large Steiner systems always need to contain.

math.CO↗
arXiv2020

On Hamiltonian cycles in hypergraphs with dense link graphs

We show that every $k$-uniform hypergraph on $n$ vertices whose minimum $(k-2)$-degree is at least $(5/9+o(1))n^2/2$ contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and Sahueza-Matamala.

math.CO↗
arXiv2020

On local Turán problems

Since its formulation, Turán's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a $3$-uniform hypergraph $\mathcal{F}$ on $n$ vertices in which any five vertices span at least one edge, prove that $|\mathcal{F}| \ge (1/4 -o(1))\binom{n}{3}$. The construction showing that this bound would be best possible is simply $\binom{X}{3} \cup \binom{Y}{3}$ where $X$ and $Y$ evenly partition the vertex set. This construction has the following more general $(2p+1, p+1)$-property: any set of $2p+1$ vertices spans a complete sub-hypergraph on $p+1$ vertices. One of our main results says that, quite surprisingly, for all $p>2$ the $(2p+1,p+1)$-property implies the conjectured lower bound.

math.CO↗
arXiv2019

Embedding Hypertrees into Steiner Triple Systems

In this paper we are interested in the following question: Given an arbitrary Steiner triple system $S$ on $m$ vertices and any 3-uniform hypertree $T$ on $n$ vertices, is it necessary that $S$ contains $T$ as a subgraph provided $m \geq (1+μ)n$? We show the answer is positive for a class of hypertrees and conjecture that the answer is always positive.

math.CO↗
arXiv2019

Increasing paths in countable graphs

In this paper we study variations of an old result by Müller, Reiterman, and the last author stating that a countable graph has a subgraph with infinite degrees if and only if in any labeling of the vertices (or edges) of this graph by positive integers we can always find an infinite increasing path. We study corresponding questions for hypergraphs and directed graphs. For example we show that the condition that a hypergraph contains a subhypergraph with infinite degrees is equivalent to the condition that any vertex labeling contains an infinite increasing loose path. We also find an equivalent condition for a graph to have a property that any vertex labeling with positive integers contains a path of arbitrary finite length, and we study related problems for oriented graphs and labelings with $\mathbb{Z}$ (instead of $\mathbb{N}$). For example, we show that for every simple hypergraph, there is a labelling of its edges by $\mathbb{Z}$ that forbids one-way infinite increasing paths.

math.CO↗
arXiv2019

Hypergraphs with vanishing Turán density in uniformly dense hypergraphs

P. Erdős [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We study related questions for $3$-uniform hypergraphs with the additional assumption that $H$ has to be uniformly dense with respect to vertex sets. In particular, we characterise those hypergraphs $F$ that are guaranteed to appear in large uniformly dense hypergraphs $H$ of positive density. We also review the case when the density of the induced subhypergraphs of $H$ may depend on the proportion of the considered vertex sets.

math.CO↗
arXiv2017

On a Turán problem in weakly quasirandom 3-uniform hypergraphs

Extremal problems for $3$-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is weakly $(d,η)$-quasirandom if for any subset $U\subseteq V$ the number of hyperedges of $H$ contained in $U$ is in the interval $d\binom{|U|}{3}\pmη|V|^3$. We show that for any $\varepsilon>0$ there exists $η>0$ such that every sufficiently large weakly $(1/4+\varepsilon,η)$-quasirandom hypergraph contains four vertices spanning at least three hyperedges. This was conjectured by Erdős and Sós and it is known that the density $1/4$ is best possible. Recently, a computer assisted proof of this result based on the flag-algebra method was established by Glebov, Král', and Volec. In contrast to their work our proof presented here is based on the regularity method of hypergraphs and requires no heavy computations. In addition we obtain an ordered version of this result. The method of our proof allows us to study extremal problems of this type in a more systematic way and we discuss a few extensions and open problems here.

math.CO↗
arXiv2017

A Ramsey Class for Steiner Systems

We construct a Ramsey class whose objects are Steiner systems. In contrast to the situation with general $r$-uniform hypergraphs, it turns out that simply putting linear orders on their sets of vertices is not enough for this purpose: one also has to strengthen the notion of subobjects used from "induced subsystems" to something we call "strongly induced subsystems". Moreover we study the Ramsey properties of other classes of Steiner systems obtained from this class by either forgetting the order or by working with the usual notion of subsystems. This leads to a perhaps surprising induced Ramsey theorem in which designs get coloured.

math.CO↗
arXiv2017

Borsuk and Ramsey type questions in Euclidean space

We give a short survey of problems and results on (1) diameter graphs and hypergraphs, and (2) geometric Ramsey theory. We also make some modest contributions to both areas. Extending a well known theorem of Kahn and Kalai which disproved Borsuk's conjecture, we show that for any integer $r\ge 2$, there exist $\varepsilon=\varepsilon(r)>0$ and $d_0=d_0(r)$ with the following property. For every $d\ge d_0$, there is a finite point set $P\subset\mathbb{R}^d$ of diameter $1$ such that no matter how we color the elements of $P$ with fewer than $(1+\varepsilon)^{\sqrt{d}}$ colors, we can always find $r$ points of the same color, any two of which are at distance $1$.

math.CO↗
arXiv2017

Some remarks on the extremal function for uniformly two-path dense hypergraphs

We investigate extremal problems for hypergraphs satisfying the following density condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, η,P_2)$-dense if for any two subsets of pairs $P$, $Q\subseteq V\times V$ the number of pairs $((x,y),(x,z))\in P\times Q$ with $\{x,y,z\}\in E$ is at least $d|\mathcal{K}_{P_2}(P,Q)|-η|V|^3,$ where $\mathcal{K}_{P_2}(P,Q)$ denotes the set of pairs in $P\times Q$ of the form $((x,y),(x,z))$. For a given $3$-uniform hypergraph $F$ we are interested in the infimum $d\geq 0$ such that for sufficiently small $η$ every sufficiently large $(d, η,P_2)$-dense hypergraph $H$ contains a copy of $F$ and this infimum will be denoted by $π_{P_2}(F)$. We present a few results for the case when $F=K_k^{(3)}$ is a complete three uniform hypergraph on $k$ vertices. It will be shown that $π_{P_2}(K_{2^r}^{(3)})\leq \frac{r-2}{r-1}$, which is sharp for $r=2,3,4$, where the lower bound for $r=4$ is based on a result of Chung and Graham [Edge-colored complete graphs with precisely colored subgraphs, Combinatorica 3 (3-4), 315-324].

math.CO↗
arXiv2017

Ramsey Partial Orders from Acyclic Graphs

We prove that finite partial orders with a linear extension form a Ramsey class. Our proof is based on the fact that class of acyclic graphs has the Ramsey property and uses the partite construction.

math.CO↗
arXiv2017

On a generalisation of Mantel's theorem to uniformly dense hypergraphs

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem in extremal combinatorics. While for $k=2$ this problem is well understood, due to the work of Turán and of Erdős and Stone, only very little is known for $k$-uniform hypergraphs for $k>2$. We focus on the case when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices. Already this very innocent (and maybe somewhat particular looking) problem is still wide open even for $k=3$. We consider a variant of the problem where the large hypergraph $H$ enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H os and Sós about 30 years ago. We show that every $k$-uniform hypergraph $H$ with density $>2^{1-k}$ with respect to every large collections of $k$-cliques induced by sets of $(k-2)$-tuples contains a copy of $F$. The required density $2^{1-k}$ is best possible as higher order tournament constructions show. Our result can be viewed as a common generalisation of the first extremal result in graph theory due to Mantel (when $k=2$ and the hereditary density condition reduces to a normal density condition) and a recent result of Glebov, Král', and Volec (when $k=3$ and large subsets of vertices of $H$ induce a subhypergraph of density $>1/4$). Our proof for arbitrary $k\geq 2$ utilises the regularity method for hypergraphs.

math.CO↗
arXiv2016

The chromatic number of finite type-graphs

By a finite type-graph we mean a graph whose set of vertices is the set of all $k$-subsets of $[n]=\{1,2,\ldots, n\}$ for some integers $n\ge k\ge 1$, and in which two such sets are adjacent if and only if they realise a certain order type specified in advance. Examples of such graphs have been investigated in a great variety of contexts in the literature with particular attention being paid to their chromatic number. In recent joint work with Tomasz Łuczak, two of the authors embarked on a systematic study of the chromatic numbers of such type-graphs, formulated a general conjecture determining this number up to a multiplicative factor, and proved various results of this kind. In this article we fully prove this conjecture.

math.CO↗