An application of Lovász' local lemma-A new lower bound for the van der Waerden number

1990 ◽  
Vol 1 (3) ◽  
pp. 343-360 ◽  
Author(s):  
Zoltán Szabó
Keyword(s):  
Lower Bound ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Lovasz Local Lemma

scholarly journals A lower bound for the distributed Lovász local lemma

2016 ◽  
Author(s):  
Sebastian Brandt ◽  
Orr Fischer ◽  
Juho Hirvonen ◽  
Barbara Keller ◽  
Tuomo Lempiäinen ◽  
...  
Keyword(s):  
Lower Bound ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Lovasz Local Lemma

scholarly journals Lower Bounds on van der Waerden Numbers: Randomized- and Deterministic-Constructive

10.37236/551 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
William Gasarch ◽  
Bernhard Haeupler
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Efficient Algorithm ◽  
Probabilistic Method ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Constructive Proofs ◽  
Different Types ◽  
Lovasz Local Lemma

The van der Waerden number $W(k,2)$ is the smallest integer $n$ such that every $2$-coloring of 1 to $n$ has a monochromatic arithmetic progression of length $k$. The existence of such an $n$ for any $k$ is due to van der Waerden but known upper bounds on $W(k,2)$ are enormous. Much effort was put into developing lower bounds on $W(k,2)$. Most of these lower bound proofs employ the probabilistic method often in combination with the Lovász Local Lemma. While these proofs show the existence of a $2$-coloring that has no monochromatic arithmetic progression of length $k$ they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm. This paper clarifies these notions and gives definitions for deterministic- and randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on $W(k,2)$ in this light. We show how known nonconstructive lower bound proofs based on the Lovász Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms.

scholarly journals Can Colour-Blind Distinguish Colour Palettes?

10.37236/2319 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Large Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Comprehensive Information ◽  
Lovasz Local Lemma ◽  
Delta G ◽  
Irregular Graphs

Let $c:E(G)\rightarrow [k]$ be  a colouring, not necessarily proper, of edges of a graph $G$. For a vertex $v\in V$, let $\overline{c}(v)=(a_1,\ldots,a_k)$, where $ a_i =|\{u:uv\in E(G),\;c(uv)=i\}|$, for $i\in [k].$ If we re-order the sequence $\overline{c}(v)$ non-decreasingly, we obtain a sequence $c^*(v)=(d_1,\ldots,d_k)$, called a palette of a vertex $v$. This can be viewed as the most comprehensive information about colours incident with $v$ which can be delivered by a person who is unable to name colours but distinguishes one from another. The smallest $k$ such that $c^*$ is a proper colouring of vertices of $G$ is called the colour-blind index of a graph $G$, and is denoted by dal$(G)$. We conjecture that there is a constant $K$ such that dal$(G)\leq K$ for every graph $G$ for which the parameter is well defined. As our main result we prove that $K\leq 6$ for regular graphs of sufficiently large degree, and for irregular graphs with $\delta (G)$ and $\Delta(G)$ satisfying certain conditions. The proofs are based on the Lopsided Lovász Local Lemma. We also show that $K=3$ for all regular bipartite graphs, and for complete graphs of order $n\geq 8$.

An Alternate Proof of the Algorithmic Lovász Local Lemma

2017 ◽  
pp. 61-65
Author(s):  
Ioannis Giotis ◽  
Lefteris Kirousis ◽  
Kostas I. Psaromiligkos ◽  
Dimitrios M. Thilikos
Keyword(s):  
Lovász Local Lemma ◽  
Alternate Proof ◽  
Local Lemma ◽  
Lovasz Local Lemma

scholarly journals Uniform Sampling Through the Lovász Local Lemma

2019 ◽  
Vol 66 (3) ◽  
pp. 1-31 ◽  
Author(s):  
Heng Guo ◽  
Mark Jerrum ◽  
Jingcheng Liu
Keyword(s):  
Lovász Local Lemma ◽  
Uniform Sampling ◽  
Local Lemma ◽  
Lovasz Local Lemma

scholarly journals Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma

2017 ◽  
Vol 49 (1) ◽  
pp. 1-23
Author(s):  
Christoph Hofer-Temmel
Keyword(s):  
Point Process ◽  
Lower Bounds ◽  
Hard Sphere ◽  
Hard Core ◽  
Combinatorial Structure ◽  
Hard Sphere Model ◽  
Sphere Model ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Lovasz Local Lemma

AbstractA point process isR-dependent if it behaves independently beyond the minimum distanceR. In this paper we investigate uniform positive lower bounds on the avoidance functions ofR-dependent simple point processes with a common intensity. Intensities with such bounds are characterised by the existence of Shearer's point process, the uniqueR-dependent andR-hard-core point process with a given intensity. We also present several extensions of the Lovász local lemma, a sufficient condition on the intensity andRto guarantee the existence of Shearer's point process and exponential lower bounds. Shearer's point process shares a combinatorial structure with the hard-sphere model with radiusR, the uniqueR-hard-core Markov point process. Bounds from the Lovász local lemma convert into lower bounds on the radius of convergence of a high-temperature cluster expansion of the hard-sphere model. This recovers a classic result of Ruelle (1969) on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive approach of Dobrushin (1996).

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