arXiv subjects
Publications and source records attributed to Vojtěch Rödl.
We extend the hypergraph regularity method to sparse $3$-uniform hypergraphs whose vertex links are $C_4$-free. In other words, we consider hypergraphs $H=(V,E)$ that are $K_{1,2,2}$-free, which implies that $|E|=O(|V|^{5/2})$. For such hypergraphs we establish a sparse analogue of the removal lemma for the tight cycle on seven vertices minus an edge.
Extending earlier results of Nešetřil and Rödl [Selective graphs and hypergraphs, Ann. Discrete Math. 3 (1978), 181--189], we show that for every ordered graph $F$ there exist an ordered graph $H$ and a system $\mathscr{H}_F$ of induced copies of $F$ such that every colouring of the edges of $H$ yields a canonically coloured copy of $F$ from $\mathscr{H}_F$ and any two copies from $\mathscr{H}_F$ intersect either in a vertex or an edge or not at all. As a consequence, this allows us to construct, for any given ordered graph $F$, canonical Ramsey graphs $H$ enjoying additional structural properties. In particular, $H$ can have the same clique number as $F$ and, provided $F$ is not bipartite, the same odd girth. Moreover, if $F$ is connected, then the copies of $F$ from $\mathscr{H}_F$ are not only induced, but their pairs of vertices also have the same distances in $H$ as in $F$.
We will prove that $R_k(k+1,k+1)\geq 4 tw_{\lfloor k/4\rfloor -3}(2)$, where $tw$ is the tower function defined by ${tw}_1(x)=x$ and ${tw}_{i+1}(x)=2^{{tw}_i(x)}$. We also give proofs of $R_k(k+1,k+2)\geq 4 tw_{k-7}(2)$, $R_k(k+1,2k+1)\geq 4 tw_{k-3}(2)$, and $R_k(k+2,k+2)\geq 4 tw_{k-4}(2)$.
A $k$-uniform hypergraph $H$ is called a partial $(k,\ell)$-system if every set of $\ell$ vertices of $V(H)$ is contained in at most one edge of $H$. We prove the existence of a partial $(k,k-1)$-system $H$ whose Ramsey number with $r \geq 4$ colors grows as a tower of height $k-1$.
For $0<δ\leq 1$, let $R_k(m;δ)$ denote the smallest $N$ such that every coloring of $k$-element subsets by two colors yields an $m$-element set $M$ with relative discrepancy $δ$, which means that one color class has at least $(\frac{1+δ}2){m\choose k}$ elements. The number $R_k(m;δ)$ may be viewed as an extension of the usual $k$-hypergraph Ramsey number because $R_k(m)=R_k(m,1)$. Our main result is the following theorem. %\begin{theorem} For some constants $c,k_0$, and $\eps>0$, and for all $k\geq k_0$, $c\log k\leq n\leq k/11$, \[ R_k(k+n);2^{-\eps n})\geq \tw_{\lfloor k/n\rfloor}(2). \] %\end{theorem} In particular, for $n=\lceil c\log k\rceil$, we get a tower of height $δk/\log k$ and relative discrepancy polynomial in~$k$.
A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the area, a stronger ``density'' concept was considered in [J. Amer. Math. Soc. 3, 1--7, 1990]: If $S$ is a $d$-dimensional simplex, then for any $μ>0$ there is an integer $d:=d(S,μ)$ and finite configuration $X\subseteq \mathbb{R}^d$ such that any subconfiguration $Y\subseteq X$ with $|Y|\geq μ|X|$ contains a copy of $S$. Complementing this, here we show the existence of $μ:=μ(S)$ and of an infinite configuration $X\subseteq \mathbb{R}^{\infty}$ with the property that any finite coloring of $X$ yields a monochromatic copy of $S$, yet for any finite set of points $Y\subseteq X$ contains a subset $Z\subseteq Y$ of size $|Z|\geq μ|Y|$ without a copy of $S$.
We study subgraphs that appear in large Ramsey graphs for a given graph $F$. The recent girth Ramsey theorem of the first two authors asserts that there are Ramsey graphs such that all small subgraphs are `forests of copies of $F$' amalgamated on vertices and edges. We derive a few further consequences from this structural result and investigate to which extent such forests of copies must be present in Ramsey graphs.
We establish the existence of generalised Sidon sets enjoying additional Ramsey-type properties, which are motivated by questions of Erdős and Newman and of Alon and Erdős.
We introduce a modification of the Turán density of ordered graphs and investigate this graph parameter.
In their classical paper, Erdős, Goodman and Pósa studied the representation of a graph with vertex set $[n]$ by a family of subsets $S_1,\dots, S_n$ with the property that $\{i,j\}$ is an edge if and only if $S_i\cap S_j\neq \emptyset$. In this note, we consider a similar representation of bounded degree $r$-uniform hypergraphs and establish some bounds for a corresponding problem.
We study quantitative aspects of the following fact: For every graph $F$, there exists a graph $G$ with the property that any $2$-coloring of the triangles of $G$ yields an induced copy of $F$, in which all triangles are monochromatic. We define the Ramsey number $R_{\text{ind}}^Δ(F)$ as the smallest size of such a graph $G$. Although this fact has several proofs, all of them provide tower-type bounds. We study the number $R_{\text{ind}}^Δ(F)$ for some particular classes of graphs $F$.
We construct for every integer $k\geq 3$ and every real $μ\in(0, \frac{k-1}{k})$ a set of integers $X=X(k, μ)$ which, when coloured with finitely many colours, contains a monochromatic $k$-term arithmetic progression, whilst every finite $Y\subseteq X$ has a subset $Z\subseteq Y$ of size $|Z|\geq μ|Y|$ that is free of arithmetic progressions of length $k$. This answers a question of Erdős, Nešetřil, and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett version of this result.
Daisies are a special type of hypergraphs introduced by Bollobás, Leader and Malvenuto. An $r$-daisy determined by a pair of disjoint sets $K$ and $M$ is the $(r+|K|)$-uniform hypergraph $\{K\cup P:\: P\in M^{(r)}\}$. In [Combin. Probab. Comput. 20, no. 5, 743-747, 2011] the authors studied Turán type density problems for daisies. This paper deals with Ramsey numbers of Daisies, which are natural generalizations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of $r$-daisies and also for special cases where the size of the kernel is bounded.
We show that $3$-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than $1/3$ contain a clique on five vertices. This result is asymptotically best possible.
The $k$-representation number of a graph $G$ is the minimum cardinality of the system of vertex subsets with the property that every edge of $G$ is covered at least $k$ times while every non-edge is covered at most $(k-1)$ times. In particular, for $k=1$ this notion is equivalent to the clique number of a graph $G$. Extending results of Frieze and Reed, and Eaton and Grable, we study the $k$-representation number of $G(n,1/2)$. As a tool, we will prove a sharp concentration result counting the number of induced subgraphs of $G(n,1/2)$ with density $(\frac{1}{2}+α)$. In Lemma 3.7, we will show that the number of such subgraphs is close to its expected value with probability $1-\exp(-n^C)$.
Given a hypergraph $F$ and a number of colours $r$, there exists a hypergraph $H$ of the same girth satisfying $H\longrightarrow (F)_r$. Moreover, for every linear hypergraph $F$ there exists a Ramsey hypergraph $H$ that locally looks like a forest of copies of $F$.
A well-known result of Ajtai et al. from 1982 states that every $k$-graph $H$ on $n$ vertices, with girth at least five, and average degree $t^{k-1}$ contains an independent set of size $c n (\log t)^{1/(k-1)}/t$ for some $c>0$. In this paper we show that an independent set of the same size can be found under weaker conditions allowing certain cycles of length 2, 3 and 4. Our work is motivated by a problem of Lo and Zhao, who asked for $k\ge 4$, how large of an independent set a $k$-graph $H$ on $n$ vertices necessarily has when its maximum $(k-2)$-degree $Δ_{k-2}(H)\le dn$. (The corresponding problem with respect to $(k-1)$-degrees was solved by Kostochka, Mubayi, and Varstraëte [Random Structures & Algorithms 44, 224--239, 2014].) In this paper we show that every $k$-graph $H$ on $n$ vertices with $Δ_{k-2}(H)\le dn$ contains an independent set of size $c (\frac nd \log\log \frac nd)^{1/(k-1)}$, and under additional conditions, an independent set of size $c (\frac nd \log \frac nd)^{1/(k-1)}$. The former assertion gives a new upper bound for the $(k-2)$-degree Turán density of complete $k$-graphs.
For integers $k\ge 2$ and $N\ge 2k+1$ there are $k!2^k$ canonical orderings of the edges of the complete $k$-uniform hypergraph with vertex set $[N] = \{1,2,\dots, N\}$. These are exactly the orderings with the property that any two subsets $A, B\subseteq [N]$ of the same size induce isomorphic suborderings. We study the associated canonisation problem to estimate, given $k$ and $n$, the least integer $N$ such that no matter how the $k$-subsets of $[N]$ are ordered there always exists an $n$-element set $X\subseteq [N]$ whose $k$-subsets are ordered canonically. For fixed $k$ we prove lower and upper bounds on these numbers that are $k$ times iterated exponential in a polynomial of $n$.