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

Massimo Lauria, Pavel Pudlák, Vojtěch Rödl, Neil Thapen
2013 arXiv   pre-print
We show 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

Massimo Lauria, Pavel Pudlák, Vojtěch Rödl, Neil Thapen
2016 Combinatorica
We show 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]

Massimo Lauria, Pavel Pudlák, Vojtěch Rödl, Neil Thapen
2013 Lecture Notes in Computer Science
We show 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]

Alexander Engstrom
2010 arXiv   pre-print
We show 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  .  ...

### 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

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.  ...  Relations of Ramsey-type theorems to various fields in mathematics are well documented in published books and monographs.  ...  Slany  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]

Natasha Dobrinen
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  ) 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  . 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]

Ron L. Graham, Jaroslav Nešetřil
2013 The Mathematics of Paul Erdős II
But perhaps one could say 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]

R. L. Graham, J. Nešetřil
1997 Algorithms and Combinatorics
But perhaps one could say 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]

Marthe Bonamy, Michał Pilipczuk
2020 arXiv   pre-print
We 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

Ronald de Haan, Jakub Szymanik
2019 Mathematical Structures in Computer Science
In this paper, we first show 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]

István Tomon
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  , 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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