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A solution to a colouring problem of P. Erdős

1992
*
Discrete Mathematics
*

Introduction The purpose

doi:10.1016/0012-365x(92)90588-7
fatcat:ewc5exqhrracfjhahvmo4s7p3q
*of*this paper is*to*prove Theorem 1.1 which provides an affirmative*solution**to**a**colouring**problem*posed by*P*. ... For*a*given graph G, x(G) is the minimum number*of**colours*needed*to**colour*the vertices*of*G such that adjacent vertices have distinct*colours*. ...##
###
The Erdös-Lovász Tihany conjecture and complete minors

2011
*
Journal of Combinatorics
*

In this paper we settle this latter

doi:10.4310/joc.2011.v2.n4.a6
fatcat:acsfal4wbrcpbglfkrzakz2wri
*problem*for*a*few small additional values*of*s and t. ... If Hadwiger's Conjecture holds, then this latter*problem*might be easier*to*settle than the*Erdős*-Lovász Tihany Conjecture. ... Nevertheless, we expect*Problem*3.1*to*be very difficult, since*a*positive*solution**of**Problem*3.1 restricted*to*6-chromatic graphs would imply*a*positive*solution**to*the Double-Critical Graph Conjecture ...##
###
Embedding graphs into larger graphs: results, methods, and problems
[article]

2019
*
arXiv
*
pre-print

Extremal Graph Theory is

arXiv:1912.02068v1
fatcat:2rgpm6wbmvafta74cmcbbxhcs4
*a*very deep and wide area*of*modern combinatorics. ... Some results discussed here got stronger emphasis, since they are connected*to*Lov\'asz (and sometimes*to*us). ... We thank above all*to*József Balogh and András Gyárfás, and also*to*Zoltán Füredi, János Pach, Jan Hladký, Zoltán L. ...##
###
A Parallel Branch and Bound Algorithm for the Maximum Labelled Clique Problem
[article]

2014
*
arXiv
*
pre-print

The maximum labelled clique

arXiv:1407.7061v2
fatcat:4qmybq2nevd3pjvjmbrdpbteyi
*problem*is*a*variant*of*the maximum clique*problem*where edges in the graph are given labels, and we are not allowed*to*use more than*a*certain number*of*distinct labels in ...*a**solution*. ... and from*P*22 colourOrder :: (Vertex Set*P*) → (Vertex Array, Int Array) 23 begin 24 (order , bounds ) ← ([],*colourable*= ∅ do 30 v ← the first vertex*of**colourable*31 append v*to*order , and*colour**to*...##
###
A parallel branch and bound algorithm for the maximum labelled clique problem

2014
*
Optimization Letters
*

The maximum labelled clique

doi:10.1007/s11590-014-0837-4
fatcat:ngot4ggnxfaejbim5tq2u3xzei
*problem*is*a*variant*of*the maximum clique*problem*where edges in the graph are given labels, and we are not allowed*to*use more than*a*certain number*of*distinct labels in ...*a**solution*. ... Open Access This article is distributed under the terms*of*the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) ...##
###
Some solved and unsolved problems in combinatorial number theory, ii

1993
*
Colloquium Mathematicum
*

called

doi:10.4064/cm-65-2-201-211
fatcat:a5d7mo5dcvfennvvsb2acdgh7e
*a*k-partition (or k-*colouring*)*of*S, and the subsets*A*1 , . . . ,*A*k are referred*to*as classes. ... integer, then let f (N,*A*, t) denote the number*of**solutions**of*N i=1 ε i*a*i = t where ε i = 0 or 1 . ...##
###
On zero-sum turan problems of Bialostocki and Dierker

1992
*
Journal of the Australian Mathematical Society
*

*A*Z k -

*colouring*

*of*

*a*graph G is

*a*

*colouring*

*of*the edges

*of*G by Z k , the additive group

*of*integers modulo k , avoiding

*a*copy

*of*

*a*given graph H, for which the sum

*of*the values on its edges is 0 ( ... By the Zero-Sum Turan number, denoted T(n, G, Z k ), k | m , we mean the maximum number

*of*edges in

*a*Z k -

*colouring*

*of*

*a*graph on n vertices that contains no zero-sum (mod k) copy

*of*G . ... Acknowledgement We would like

*to*thank the referee for his helpful remarks. On zero-sum

*problems*...

##
###
Paul Erdős's (1913-1996)

1997
*
Acta Arithmetica
*

As an answer

doi:10.4064/aa-81-4-299-317
fatcat:ofairefcnjar7d4gtbmcsg6l4m
*to*this question, I soon published (in the Acta Arithmetica) my first paper based on an*Erdős**problem*. ... However, when after*a*good sleep you awoke next morning your head was full*of*his*problems*and ideas, and you were just unable*to*think about anything else; and in at most two days you were longing again ... for the*solution**of**a*numbertheoretic*problem*. ...##
###
A combinatorial mathematician in Norway: some personal reflections

2001
*
Discrete Mathematics
*

In the paper, I ÿrst try

doi:10.1016/s0012-365x(01)00222-9
fatcat:thxl5gyhlvedxocgf2qr2eno4e
*to*give some impression*of*Norwegian contributions*to*combinatorics in the 20th century. This is followed by some remarks on my own combinatorial experiences. ... Later, I was*to*meet him quite often at conferences. I even checked*a**solution*by David Preiss*of*one*of**Erdős*' prize*problems*when we were in Durham, UK, in 1974. ... Secondly, he notes that for*p*≡ 1 (mod 4) there are*solutions**of*the congru-ence*a*2 ≡ −1 (mod*p*), as follows from the well-known fact-Wilson's theorem-that (*p*− 1)! ≡ −1 (mod*p*) for all primes*p*. ...##
###
Solved and Unsolved Problems

2017
*
EMS Newsletter
*

I Six new

doi:10.4171/news/105/14
fatcat:bfda2dankbeijot2mwmpgjat4m
*problems*-*solutions*solicited*Solutions*will appear in*a*subsequent issue. 179. Let*p*=*p*1*p*2 · · ·*p*n and q = q 1 q 2 · · · q n be two permutations. ... Does an n-universal word*of*length n 2 − 2n + 4 exist? 185* (*Erdős*' unit distance*problem*). ... Therefore We also solicit your new*problems*with their*solutions*for the next "Solved and Unsolved*Problems*" column, which will be devoted*to*Fundamentals*of*Mathematical Analysis. ...##
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The Erdős–Rothschild problem on edge-colourings with forbidden monochromatic cliques

2017
*
Mathematical proceedings of the Cambridge Philosophical Society (Print)
*

Our final result is

doi:10.1017/s0305004116001031
fatcat:3gpwm5gty5evpom2qryiitznti
*a*stability theorem for complete multipartite graphs G, describing the asymptotic structure*of*such G with F(G; k ) = F(n; k ) · 2 o(n 2) in terms*of**solutions**to*the optimisation*problem*... This*problem*was first considered by*Erdős*and Rothschild in 1974, but it has been solved only for*a*very small number*of*non-trivial cases. ...*Erdős*and Rothschild also considered the generalisation*of*the*problem*, where one forbids*a*monochromatic graph H (the same for each*colour*). ...##
###
Clique Is Hard on Average for Regular Resolution
[article]

2020
*
arXiv
*
pre-print

We prove that for k ≪√(n) regular resolution requires length n^Ω(k)

arXiv:2012.09476v1
fatcat:3agob6qnf5btnp6ccd5i4smwgq
*to*establish that an*Erdős*-Rényi graph with appropriately chosen edge density does not contain*a*k-clique. ... This lower bound is optimal up*to*the multiplicative constant in the exponent, and also implies unconditional n^Ω(k) lower bounds on running time for several state-*of*-the-art algorithms for finding maximum ... Acknowledgements This work has been*a*long journey, and different subsets*of*the authors want*to*acknowledge fruitful and enlightening discussions with different subsets from the following list*of*colleagues ...##
###
Sums and products along sparse graphs
[article]

2009
*
arXiv
*
pre-print

n^1/2+δ for its sum-product over the integers implies

arXiv:0905.0135v4
fatcat:ueztglxy3bebzjxrw2bfkq73ju
*a*lower bound*of*n^1+δ for the original*Erdős*-Szemerédi*problem*. ...*A*key element in our proofs is*a*reduction from the sum-product*of**a*matching*to*the maximum number*of*translates*of**a*set*of*integers into the perfect squares. ... This work was initiated while the second and third authors were visiting the Theory Group*of*Microsoft Research. ...##
###
Paul Erdos. 26 March 1913 -- 20 September 1996: Elected For.Mem.R.S. 1989

1999
*
Biographical Memoirs of Fellows of the Royal Society
*

Szeg ,

doi:10.1098/rsbm.1999.0011
fatcat:4skirt4ha5bilhz63yusrz62h4
*P*. Turán and perhaps, above all, the subject*of*this memoir, Pál (Paul)*Erdos*. ... As Ernst Straus put it,*Erdos*was 'the crown prince*of**problem*solvers and the undisputed monarch*of**problem*posers'. ... logn 1 2 n + o(n) f (m) ∑*p*f (*p*) 2 /*p*= ∞ x ∈r lim n→ ∞*A*x (n)/n = F (x), f ′(*p*) = f (*p*) | f (*p*)| ≤ 1 f ′(*p*) = 1 ∑ f (*p*)≠0 1/*p**A**problem**of**a*rather different nature also occupied*Erdos*for several ...##
###
Structural and Logical Approaches to the Graph Isomorphism Problem
[chapter]

2012
*
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
*

*To*use

*colour*refinement as an isomorphism test, apply it

*to*disjoint union

*of*the input graphs G , H.Colour refinement distinguishes G and H if there is

*a*

*colour*c

*of*the stable

*colouring*such that G ... Theorem (Tinhofer 1991)

*Colour*refinement distinguishes G and H iff the linear program has no

*solution*. ... Level-k Sherali-Adams relaxation (L k ) Variables Otto 2012) k-WL distinguishes G and H iff COMP k−1 ∪ CONT k ∪ NN k has

*a*

*solution*. ...

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