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The complexity of proving that a graph is Ramsey
[article]

2013
*
arXiv
*
pre-print

We show

arXiv:1303.3166v1
fatcat:f3sffmoqozhthagv4cplmoijxe
*a*superpolynomial lower bound on*the*length*of*resolution proofs*that*G*is*c-*Ramsey*, for every*graph*G. ... Our proof makes use*of**the*fact*that*every*Ramsey**graph*must contain*a*large subgraph with some*of**the*statistical properties*of**the*random*graph*. ...*A*c-*Ramsey**graph**is**a*witness*that*r(c log n) > n, so*proving**that**a**graph**is**Ramsey**is*in some sense*proving**a*lower bound for r(k). ...##
###
The complexity of proving that a graph is Ramsey

2016
*
Combinatorica
*

We show

doi:10.1007/s00493-015-3193-9
fatcat:xtmpehkseba7bh3xup3z6cua2e
*a*superpolynomial lower bound on*the*length*of*resolution proofs*that*G*is*c-*Ramsey*, for every*graph*G. ... Our proof makes use*of**the*fact*that*every c-*Ramsey**graph*must contain*a*large subgraph with some properties typical for random*graphs*. ... Since*a*c-*Ramsey**graph**is**a*witness*that*r(c log n) > n,*proving**that**a**graph**is*c-*Ramsey**is*, in some sense,*proving**a*lower bound for r(k). ...##
###
The Complexity of Proving That a Graph Is Ramsey
[chapter]

2013
*
Lecture Notes in Computer Science
*

We show

doi:10.1007/978-3-642-39206-1_58
fatcat:jpfhq2676fhsvb2li5wi2o662q
*a*superpolynomial lower bound on*the*length*of*resolution proofs*that*G*is*c-*Ramsey*, for every*graph*G. ... Our proof makes use*of**the*fact*that*every c-*Ramsey**graph*must contain*a*large subgraph with some properties typical for random*graphs*. ... Since*a*c-*Ramsey**graph**is**a*witness*that*r(c log n) > n,*proving**that**a**graph**is*c-*Ramsey**is*, in some sense,*proving**a*lower bound for r(k). ...##
###
Cohomological Ramsey Theory
[article]

2010
*
arXiv
*
pre-print

We show

arXiv:1002.4074v1
fatcat:cxsuvleberfdzc7fkajwnjqz3m
*that**the*vanishing*of*certain cohomology groups*of*polyhedral*complexes*imply upper bounds on*Ramsey*numbers. Lovasz bounded*the*chromatic numbers*of**graphs*using Hom*complexes*. ... Babson and Kozlov*proved*Lovasz conjecture and developed*a*Hom*complex*theory. We generalize*the*Hom*complexes*to*Ramsey**complexes*. ... Lovász*proved**that**the*chromatic number*of**a**graph*can be bounded by*the*connectivity*of*polyhedral*complexes*[8] . ...##
###
Page 65 of Mathematical Reviews Vol. , Issue 2002A
[page]

2002
*
Mathematical Reviews
*

We

*prove**that**the*recognition*of*disk-intersection*graphs*(in*the*unbounded ratio case)*is*NP- hard. This result*is**proved*in*a*more general setting*of*noncrossing arc-connected sets. ... In particular, we*prove**that**the*recognition*of*unit-ball contact*graphs**is*NP-hard in dimensions 3, 4, 8 and 24.” lhe article contains interesting constructions which may be used in studying*the**complexity*...##
###
Ramsey Theory Applications

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. ... Relations

*of*

*Ramsey*-type theorems to various fields in mathematics are well documented in published books and monographs. ... Slany [245] studied in general

*the*

*complexity*

*of*

*graph*

*Ramsey*games and

*proved*

*that*

*the*achievement game and several variants are PSPACE-complete [245, 244] , where PSPACE

*is*

*the*class

*of*problems

*that*...

##
###
Ramsey Theory on Trees and Applications
[chapter]

2016
*
Lecture Notes in Computer Science
*

*A*key result en route to

*the*proof

*that*

*the*Boolean Prime Ideal Theorem

*is*strictly weaker than

*the*Axiom

*of*Choice (see [9] )

*is*

*the*

*Ramsey*-type theorem

*of*Halpern and Läuchli on trees. ...

*The*following

*is*

*the*Strong Tree Version

*of*

*the*Halpern-Läuchli Theorem,

*proved*in another form in [8] . Theorem 2. Let d ≥ 1 and let T i , i < d, be finitely branching trees

*of*height ω. ...

*The*author gratefully acknowledges

*the*support

*of*NSF Grants DMS-142470 and DMS-1600781. ...

##
###
Page 1493 of Mathematical Reviews Vol. , Issue 2001C
[page]

2001
*
Mathematical Reviews
*

*The*local k-

*Ramsey*{mean k-

*Ramsey*] number

*of*

*a*

*graph*G

*is*

*the*smallest order

*of*

*a*complete

*graph*for which every local [mean] k-coloring ensures

*a*monochromatic copy

*of*G. ... We

*prove*

*that*all locally Cg

*graphs*are k-divergent and

*that*

*the*diameters

*of*

*the*iterated clique

*graphs*also tend to infinity with n while

*the*sizes

*of*

*the*cliques remain bounded.” ...

##
###
Ramsey Theory in the Work of Paul Erdős
[chapter]

2013
*
The Mathematics of Paul Erdős II
*

But perhaps one could say

doi:10.1007/978-1-4614-7254-4_13
fatcat:vqvyhfhj3ngr7c3vokrsmrt2ai
*that**Ramsey*theory was created largely by him. This paper will attempt to demonstrate this claim. ... W e s*a*y*that**a*class K*of**graphs**is**Ramsey*if for every choice*of*ordered*graphs**A*; B from K there exists C 2 K such*that*C ! B*A*2 . Here,*the*notation C ! ... It*is*also*proved**that*h2k + 1 ; c 1+1=k and this*is**proved*by*a*variant*of**the*greedy algorithm by induction on`. ...##
###
Ramsey Theory in the Work of Paul Erdős
[chapter]

1997
*
Algorithms and Combinatorics
*

But perhaps one could say

doi:10.1007/978-3-642-60406-5_16
fatcat:cqkc3inzjfe4lbu2jj4oem2fv4
*that**Ramsey*theory was created largely by him. This paper will attempt to demonstrate this claim. ... W e s*a*y*that**a*class K*of**graphs**is**Ramsey*if for every choice*of*ordered*graphs**A*; B from K there exists C 2 K such*that*C ! B*A*2 . Here,*the*notation C ! ... It*is*also*proved**that*h2k + 1 ; c 1+1=k and this*is**proved*by*a*variant*of**the*greedy algorithm by induction on`. ...##
###
Page 6628 of Mathematical Reviews Vol. , Issue 99j
[page]

1999
*
Mathematical Reviews
*

Summary: “

*A*corrected proof*is*given for*the*existence*of**a*universal countable {C3, Cs,---, C2;41}-free*graph*. We also*prove**that*there*is**a*universal countable ><-free*graph*. ... “This paper*is*organised as follows: It begins by exploring aspects*of*structural*Ramsey*theory.*The*focus*is*on*the*idea*of**a**Ramsey*object. ...##
###
Graphs of bounded cliquewidth are polynomially χ-bounded
[article]

2020
*
arXiv
*
pre-print

We

arXiv:1910.00697v3
fatcat:nufdhra6trfbhho646engvt5e4
*prove**that*if C*is**a*hereditary class*of**graphs**that**is*polynomially χ-bounded, then*the*class*of**graphs**that*admit decompositions into pieces belonging to C along cuts*of*bounded rank*is*also polynomially ... In particular, this implies*that*for every positive integer k,*the*class*of**graphs**of*cliquewidth at most k*is*polynomially χ-bounded. ... Acknowledgments We would like to express our gratitude to Rose McCarty for communicating*the*problem to us and for bringing*the*works*of*Chudnovsky et al. [CPST13] and*of*Kim et al. ...##
###
Page 16 of Mathematical Reviews Vol. , Issue 90M
[page]

1990
*
Mathematical Reviews
*

They

*prove**that*for every natural number / and positive constant c < 1// there*is*an no(/,c) such*that*for all n > mo and k < exp(c,/logn/2), every*graph**of*order n either contains*a*k-element homogeneous ... Let ¥ and # denote families*of**graphs*.*The**Ramsey*number r(¥,#)*is**the*smallest p for which every (red, blue) coloring*of**the*edges*of*K, yields*a*red copy*of*GE or else*a*blue copy*of*H € #. ...##
###
Characterizing polynomial Ramsey quantifiers

2019
*
Mathematical Structures in Computer Science
*

In this paper, we first show

doi:10.1017/s0960129518000397
fatcat:2itdqazrobdyvblng35xspsb3e
*that*there exist intermediate*Ramsey*quantifiers and then we*prove**a*dichotomy result for*a*large and natural class*of**Ramsey*quantifiers, based on*a*reasonable and widely ... We show*that**the*polynomial-time computable quantifiers in this class are exactly*the*constant-log-bounded*Ramsey*quantifiers. ...*The*following result,*that*we will use to*prove**the*existence*of*intermediate*Ramsey*quantifiers (assuming*the*ETH)*is*an example*of**a*lower bound based on*the*ETH. ...##
###
Ramsey properties of semilinear graphs
[article]

2021
*
arXiv
*
pre-print

*That*

*is*,

*the*exponent

*of*n does not grow with

*the*dimension. We

*prove*

*a*result about

*the*symmetric

*Ramsey*properties

*of*semilinear

*graphs*, which puts this phenomenon in

*a*more general context. ... More precisely, we

*prove*

*that*if G

*is*

*a*semilinear

*graph*

*of*

*complexity*t which contains no clique

*of*size s and no independent set

*of*size n, then G has at most O_s,t(n)·(log n)^O_t(1) vertices. ... Asymmetric

*Ramsey*properties

*of*semilinear

*graphs*In [5] , it was

*proved*

*that*if G

*is*

*a*semilinear

*graph*

*of*

*complexity*t on n vertices, and G does not contain

*the*complete bipartite

*graph*K k,k , then ...

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