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ACKNOWLEDGMENTS We thank Olivier Riordan for suggesting the proof of Theorem 1* and Vojtěch Rödl for his insightful remarks and comments. VERTEX RAMSEY PROPERTIES ... Thus, every finite family of graphs G 1 , ..., G r can be associated with a (vertex) Ramsey property. ... L FINAL REMARKS The most interesting open question concerning vertex Ramsey properties of families of graphs is whether for every two families G and H of connected nontrivial graphs the condition (G) ...doi:10.1006/jctb.2001.2078 fatcat:3dji2chtk5dvjngnpnxlfeluji
We say that G is a (3, 3)-Ramsey graph if every 2-coloring of the edges of G forces a monochromatic triangle. ... In this work we find all minimal (3, 3)-Ramsey graphs with up to 13 vertices with the help of a computer, and we obtain some new results for these graphs. ... Nedyalko Nenov, who read the manuscript and made some suggestions which led to the improvement of this work. ...arXiv:1604.03716v1 fatcat:j4v5x4mb6nev7gykuoswpj5xwe
A graph G is called (H ) k -vertex Ramsey minimal if G v → (H ) k , but G v 9 (H ) k for any proper subgraph G of G. The class of (P3) k -vertex Ramsey minimal graphs is investigated. ... Let P be a property of graphs. ... Acknowledgements The authors are very grateful to the referees for their help in preparing the ÿnal version of the paper. ...doi:10.1016/j.disc.2003.11.042 fatcat:ewtmvbyyubhnllr7loayugptfe
New families of intersection graphs of filaments are defined using families of circular arcs of a circle and families of subtrees of a tree or of a cactus.” ... Summary: “We describe a method of defining new families of graphs called G-mixed, using the method of partitioning the edge set of overlap graphs. Consider a hereditary family G of graphs. ...
colored complete graph K lex p is a Ramsey family. ... The second version of the Canonical Ramsey Theorem above says that the three-pattern family consisting of a monochromatic complete graph K mono p , a polychromatic complete graph K poly p , and a lexically ...doi:10.1007/s00373-004-0562-y fatcat:kwozlhqgrffe5l5cvaralz6you
colored complete graph K lex p is a Ramsey family. ... The second version of the Canonical Ramsey Theorem above says that the three-pattern family consisting of a monochromatic complete graph K mono p , a polychromatic complete graph K poly p , and a lexically ...doi:10.1002/jgt.10072 fatcat:iuikbucfjbavxfkn52ehvkd46e
us information on the vertex Ramsey properties of typical graphs of a given density. ... Moreover, we will need to appeal to the Ramsey properties of the random graph over various vertex subsets. ... the bounds, it would help to better understand the k-covers of a graph F, and to see if there are different colorings of G n ∪ G(n, p) that show (4.3) is not tight. ...doi:10.1002/rsa.20971 fatcat:uaqqqfe6gbfpnaq3mo5vl3tjmq
The author gratefully acknowledges the support of the Canadian National Research Council in the form of research Grant A7544. ... Let G be an r-graph, P be a hereditary property (see [8, p. 96] ) and for v e V(G) let SP P (v) be a family of subgraphs of G with the following properties: (SL)HG^P(V)^>VEV(H). ... of any vertex from Wi restores the property P. ...doi:10.4153/cmb-1972-063-2 fatcat:b55n2b3zvrb7tjqpthbyko6aiu
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
Forb(G) is the family of graphs H such that there is no embedding from G into H. A family ¥ of graphs is Ramsey if for all graphs B € ¥ there is a graph A €.F such that A -—(B)!. ... indi- cates that H is vertex Ramsey to G, that is, any vertex coloring of H with r colors admits a monochromatic induced copy of G. ...
A graph G is Ramsey for a graph H if every colouring of the edges of G in two colours contains a monochromatic copy of H. ... Two graphs H_1 and H_2 are Ramsey equivalent if any graph G is Ramsey for H_1 if and only if it is Ramsey for H_2. ... Acknowledgements The author would like to thank the University of Cambridge Faculty of Mathematics Summer Research Scheme and the Lister Fund of Queens' College, Cambridge for funding this work, Julia ...arXiv:1909.02590v1 fatcat:gpzfz3b3dvcftb67mukk4tfi7u
In this paper, the authors show that the Ramsey property holds simultaneously for arbitrary collections of Ramsey families under quite general conditions. ... A family ¥ of finite subsets of a set U is said to be Ramsey if when U is partitioned into finitely many parts, there is a part which contains an F €§ as a subset. ...
of families of graphs. ... Gabriel Semanisin (SK-SFRKS-GA; KoSice) 2003k:05084 05C55 Luczak, Tomasz (PL-POZNM-DM; Poznan); Rucinski, Andrzej (PL-POZNM-DM; Poznan) ; Urbanski, Sebastian (PL-POZNM-DM; Poznan) Vertex Ramsey properties ...
The equivalence of the “Ramsey property” and the apparently weaker “pair-Ramsey property” is also shown. Ralph Faudree (1-MEMP) ... For many families of graphs, the problem of finding all graphs whose largest eigenvalue is either minimal or maximal has been solved. ...
A class of graphs J has the Ramsey property if, for every B ... For graphs A, B and a positive integer r, the relation A -• (B) l r means that whenever A is an r-colouring of the vertices of A, then there is an embedding of B into A such that A o is constant. ... A class of graphs J has the Ramsey property if, for every B G ^F, there is an A G IT such that A -> (#)£. ...doi:10.4153/cjm-1992-064-7 fatcat:vzky7za7bngnbd2gxdr2ddeyl4
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