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Independent Arithmetic Progressions in Clique-Free Graphs on the Natural Numbers

David S. Gunderson, Imre Leader, Hans Jürgen Prömel, Vojtěch Rödl
2001 Journal of combinatorial theory. Series A  
We show that if G is a K r -free graph on N, there are independent sets in G which contain an arbitrarily long arithmetic progression together with its difference.  ...  So the claim is proved in both cases, finishing the proof of the theorem. 6 Clique-free graphs on parameter words A natural approach to extend results on independent arithmetic progressions to independent  ...  Independent arithmetic progressions We start with a lemma guaranteeing independent lines in a Hales-Jewett cube on vertices of a K r -free graph.  ... 
doi:10.1006/jcta.1999.3007 fatcat:nnkob33nnrc53f4kxngbxhjz5m

Short proofs of some extremal results III [article]

David Conlon, Jacob Fox, Benny Sudakov
2020 arXiv   pre-print
These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short.  ...  We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems.  ...  We would like to thank Kevin Ford for some helpful discussions on this theme and also thank the anonymous referees for their careful reading of the paper and useful suggestions.  ... 
arXiv:1910.08661v3 fatcat:pif5dh7eynerxcabszjo2ga6fu

Combinatorial theorems relative to a random set [article]

David Conlon
2014 arXiv   pre-print
We describe recent advances in the study of random analogues of combinatorial theorems.  ...  Particular thanks are due to Wojtek Samotij for a number of detailed comments on the manuscript.  ...  I would like to thank Jacob Fox, Tim Gowers, Rob Morris, Wojtek Samotij, Mathias Schacht, Benny Sudakov and Yufei Zhao for helpful discussions on the topics in this paper.  ... 
arXiv:1404.3324v2 fatcat:tlnmtfgmizcytlsjszrqimnz4y

Independent Deuber sets in graphs on the natural numbers

David S. Gunderson, Imre Leader, Hans Jürgen Prömel, Vojtěch Rödl
2003 Journal of combinatorial theory. Series A  
Hence if G is a K k -free graph on N, then one can solve any partition regular system of equations in an independent set.  ...  We show that for any k, m, p, c, if G is a K k -free graph on N then there is an independent set of vertices in G that contains an (m, p, c)-set.  ...  The first author also gratefully acknowledges the generous hospitality of Cambridge (Trinity College), Emory, and Humboldt Universities while working on this project.  ... 
doi:10.1016/s0097-3165(03)00100-6 fatcat:2i62czbn6bcdfhiueijs3vej7m

Additive Combinatorics and its Applications in Theoretical Computer Science

Shachar Lovett
2017 Theory of Computing  
Additive combinatorics (or perhaps more accurately, arithmetic combinatorics) is a branch of mathematics which lies at the intersection of combinatorics, number theory, Fourier analysis and ergodic theory  ...  combinatorics and their applications, mainly in number theory. • A mini-course on additive combinatorics by Barak et al. [6] explores basic ideas in additive combinatorics and some of their applications  ...  A basic question in number theory is the existence of arithmetic progressions in certain sets of numbers.  ... 
doi:10.4086/toc.gs.2017.008 dblp:journals/toc/Lovett17 fatcat:z5d45zpq3vgpzk7tshdszrdwni

VERTEX COLORING OF CERTAIN DISTANCE GRAPHS

V. Yegnanarayanan, A. Parthiban
2013 International Journal of Pure and Applied Mathematics  
Fourth, we give a brief review regarding circulant graphs and highlight its importance in the computation of chromatic number of distance graphs with appropriate references.  ...  In this paper first, we give a brief introduction about integer distance graphs.  ...  p k in arithmetic progression, each p i being at most N .  ... 
doi:10.12732/ijpam.v86i4.7 fatcat:pa6zmdnj6jdq7hcmb7mb3zh2ve

Page 5986 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews  
-free graph on N, one can solve any partition regular system in an independent set. In fact, these results generalize Ramsey’s the- orem itself. Jerrold W.  ...  MB): Leader, Imre (4-LNDUC; London); Prémel, Hans Jiirgen (D-HUMB-II; Berlin) ; Rédl, Vojtéch (1-EMRY-CS; Atlanta, GA Independent Deuber sets in graphs on the natural numbers. (English summary) J.  ... 

Recent developments in graph Ramsey theory [article]

David Conlon, Jacob Fox, Benny Sudakov
2015 arXiv   pre-print
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H.  ...  Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.  ...  The authors would like to thank the anonymous referee for a number of useful comments.  ... 
arXiv:1501.02474v3 fatcat:q3qcowfhgjbp5j36oetad6qdnq

Ramsey Theory Applications

Vera Rosta
2004 Electronic Journal of Combinatorics  
The main objective of this survey is to list applications mostly in theoretical computer science of the last two decades not contained in these.  ...  There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer  ...  A graph is called Ramsey graph if both its independence number and clique number are polylogarithmic in the number of vertices. Let A be a finite Abelian group and a set K ⊆ A is symmetric if −K = K.  ... 
doi:10.37236/34 fatcat:gxrfo23hzzewjg7rez76d4xx4i

On families of weakly dependent random variables

Tomasz Łuczak
2011 Banach Center Publications  
The first one is the family K (k) of all complete k-graphs on vertices contained in [n]; the second one of more algebraic flavor is the family AP of -element arithmetic progressions contained in [n] .  ...  Suppose that we select vertices from B ⊆ A andB ⊆ [n] one by one so that we would like to minimize the final number of arithmetic progressions of length 3.  ... 
doi:10.4064/bc95-0-9 fatcat:5ugmwnlkpbgbpbkcsage5sktmu

Treewidth versus clique number. I. Graph classes with a forbidden structure [article]

Clément Dallard, Martin Milanič, Kenny Štorgel
2021 arXiv   pre-print
In addition, we propose a question about the complexity of the maximum weight independent set problem in (tw,ω)-bounded graph classes and prove that the problem is polynomial-time solvable in every class  ...  Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems.  ...  classes, to Jean-Florent Raymond for early discussions on treewidth of 1-perfectly orientable graphs, and to Bart M.  ... 
arXiv:2006.06067v3 fatcat:v2nsvungtbht3mdplyzoecfrli

On The Communication Complexity of High-Dimensional Permutations [article]

Nati Linial and and Toniann Pitassi and Adi Shraibman
2018 arXiv   pre-print
We study the multiparty communication complexity of high dimensional permutations, in the Number On the Forehead (NOF) model.  ...  Previous protocols for Exactly-n all rely on the construction of large sets of integers without a 3-term arithmetic progression.  ...  a k-term arithmetic progression.  ... 
arXiv:1706.02207v3 fatcat:xd7jh3k3jrd5jcgfmqjarl4wse

Satisfiability Allows No Nontrivial Sparsification unless the Polynomial-Time Hierarchy Collapses

Holger Dell, Dieter Van Melkebeek
2014 Journal of the ACM  
The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ ) edges unless coNP is in NP/poly, where k  ...  For the vertex cover problem on n-vertex d-uniform hypergraphs, the above statement holds for any integer d ≥ 2.  ...  We would like to thank the following people for discussions, comments, pointers to the literature, and guidance: Matt Anderson, Albert Atserias, Kord Eickmeyer, Martin  ... 
doi:10.1145/2629620 fatcat:p5rgbdgfujderlax7sxyokquay

Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses

Holger Dell, Dieter van Melkebeek
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ ) edges unless coNP is in NP/poly, where k  ...  For the vertex cover problem on n-vertex d-uniform hypergraphs, the above statement holds for any integer d ≥ 2.  ...  We would like to thank the following people for discussions, comments, pointers to the literature, and guidance: Matt Anderson, Albert Atserias, Kord Eickmeyer, Martin  ... 
doi:10.1145/1806689.1806725 dblp:conf/stoc/DellM10 fatcat:qzmziun5kjdi7dlgmexpan6tne

On the Proof Complexity of Paris-Harrington and Off-Diagonal Ramsey Tautologies

Lorenzo Carlucci, Nicola Galesi, Massimo Lauria
2016 ACM Transactions on Computational Logic  
The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2).  ...  We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial in the number of propositional variables.  ...  ACKNOWLEDGMENTS We thank the anonymous referees for comments that improved the presentation of the article.  ... 
doi:10.1145/2946801 fatcat:fqsrvvirinbtllya4juqydzf7y
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