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Vojtech Rödl

Publications and source records attributed to Vojtech Rödl.

5 recordsLinked to original sources
arXiv2021

A blurred view of Van der Waerden type theorems

Let $AP_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $ε>0$ we call a set $AP_k(ε)=\{x_0,\ldots,x_{k-1}\}$ an $ε$-approximate arithmetic progression if for some $a$ and $d$, $|x_i-(a+id)|<εd$ holds for all $i\in\{0,1\ldots,k-1\}$. Complementing earlier results of Dumitrescu, in this paper we study numerical aspects of Van der Waerden, Szemeredi and Furstenberg-Katznelson like results in which arithmetic progressions and their higher dimensional extensions are replaced by their $ε$-approximation.

math.CO↗
arXiv2019

Extractors for small zero-fixing sources

A random variable $X$ is an $(n,k)$-zero-fixing source if for some subset $V\subseteq[n]$, $X$ is the uniform distribution on the strings $\{0,1\}^n$ that are zero on every coordinate outside of $V$. An $ε$-extractor for $(n,k)$-zero-fixing sources is a mapping $F:\{0,1\}^n\to\{0,1\}^m$, for some $m$, such that $F(X)$ is $ε$-close in statistical distance to the uniform distribution on $\{0,1\}^m$ for every $(n,k)$-zero-fixing source $X$. Zero-fixing sources were introduced by Cohen and Shinkar in [10] in connection with the previously studied extractors for bit-fixing sources. They constructed, for every $μ>0$, an efficiently computable extractor that extracts a positive fraction of entropy, i.e., $Ω(k)$ bits, from $(n,k)$-zero-fixing sources where $k\geq(\log\log n)^{2+μ}$. In this paper we present two different constructions of extractors for zero-fixing sources that are able to extract a positive fraction of entropy for $k$ essentially smaller than $\log\log n$. The first extractor works for $k\geq C\log\log\log n$, for some constant $C$. The second extractor extracts a positive fraction of entropy for $k\geq \log^{(i)}n$ for any fixed $i\in \mathbb{N}$, where $\log^{(i)}$ denotes $i$-times iterated logarithm. The fraction of extracted entropy decreases with $i$. The first extractor is a function computable in polynomial time in~$n$ (for $ε=o(1)$, but not too small); the second one is computable in polynomial time when $k\leqα\log\log n/\log\log\log n$, where $α$ is a positive constant. The subject studied in this paper is closely related to Ramsey theory. We use methods developed in Ramsey theory and our results can also be interpreted as a contribution to this field.

cs.CC↗
arXiv2016

Short Proof of Erd\H os Conjecture for Triple Systems

In 1965 Erd\H os conjectured that for all $k\ge2$, $s\ge1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\F$ with $ν(\F)=s$ cannot have more than \newline $\max\{\binom{sk+k-1}k,\;\binom nk-\binom{n-s}k\}$ edges. It took almost fifty years to prove it for triple systems. In 2012 we proved the conjecture for all $s$ and all $n\ge4(s+1)$. Then Łuczak and Mieczkowska (2013) proved the conjecture for sufficiently large $s$ and all $n$. Soon after, Frankl proved it for all $s$. Here we present a simpler version of that proof which yields Erd\H os's conjecture for $s\ge33$. Our motivation is to lay down foundations for a possible proof in the much harder case $k=4$, at least for large $s$.

math.CO↗