Ramsey goodness and beyond

In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five — all but one — of the Burr-Erdős problems.

[1]  Paul Erdös,et al.  On the connection between chromatic number, maximal clique and minimal degree of a graph , 1974, Discret. Math..

[2]  Richard H. Schelp,et al.  Goodness of trees for generalized books , 1987, Graphs Comb..

[3]  M. Simonovits,et al.  Szemeredi''s Regularity Lemma and its applications in graph theory , 1995 .

[4]  Stefan A. Burr.  Multicolor Ramsey numbers involving graphs with long suspended paths , 1982, Discret. Math..

[5]  Vojtech Rödl,et al.  On graphs with small Ramsey numbers , 2001, J. Graph Theory.

[6]  Cecil C. Rousseau,et al.  A note on Ramsey numbers for books , 2005, J. Graph Theory.

[7]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[8]  Cecil C. Rousseau,et al.  Strongly regular graphs and finite Ramsey theory , 1982 .

[9]  Cecil C. Rousseau,et al.  Book Ramsey numbers. I , 2005, Random Struct. Algorithms.

[10]  B. Sudakov,et al.  Pseudo-random Graphs , 2005, math/0503745.

[11]  Stefan A. Burr,et al.  What can we hope to accomplish in generalized Ramsey theory? , 1987, Discret. Math..

[12]  Joel Friedman,et al.  A proof of Alon's second eigenvalue conjecture and related problems , 2004, ArXiv.

[13]  Béla Bollobás,et al.  Joints in graphs , 2008, Discret. Math..

[14]  Cecil C. Rousseau,et al.  Fan-complete graph Ramsey numbers , 1996, J. Graph Theory.

[15]  Richard H. Schelp,et al.  Multipartite graph—Sparse graph Ramsey numbers , 1985, Comb..

[16]  V. Rödl,et al.  On graphs with small Ramsey numbers , 2001 .

[17]  Vladimir Nikiforov.  Edge distribution of graphs with few induced copies of a given graph , 2004 .

[18]  Vladimir Nikiforov.  Edge Distribution of Graphs with Few Copies of a Given Graph , 2006, Comb. Probab. Comput..

[19]  János Komlós,et al.  Blow-up Lemma , 1997, Combinatorics, Probability and Computing.

[20]  V. Nikiforov.  Graphs with many r‐cliques have large complete r‐partite subgraphs , 2007, math/0703554.

[21]  C. C. Rousseau,et al.  The Book-Tree Ramsey Numbers , 2004 .

[22]  Cecil C. Rousseau,et al.  Large generalized books are p-good , 2004, J. Comb. Theory, Ser. B.

[23]  A. Thomason.  Pseudo-Random Graphs , 1987 .

[24]  Benny Sudakov,et al.  Large Kr‐free subgraphs in Ks‐free graphs and some other Ramsey‐type problems , 2005, Random Struct. Algorithms.

[25]  Benny Sudakov.  Large K r -free subgraphs in K s -free graphs and some other Ramsey-type problems , 2005 .

[26]  Béla Bollobás,et al.  Almost all Regular Graphs are Hamiltonian , 1983, European journal of combinatorics (Print).

[27]  S. Burr,et al.  Ramsey Numbers Involving Graphs with Long Suspended Paths , 1981 .

[28]  D. R. Lick,et al.  The Theory and Applications of Graphs. , 1983 .

[29]  Paul Erdös,et al.  Generalizations of a Ramsey-theoretic result of chvátal , 1983, J. Graph Theory.

[30]  Richard H. Schelp,et al.  On Ramsey numbers involving starlike multipartite graphs , 1983, J. Graph Theory.

[31]  Cecil C. Rousseau,et al.  A Class of Ramsey Problems Involving Trees , 1978 .

[32]  Béla Bollobás,et al.  Modern Graph Theory , 2002, Graduate Texts in Mathematics.

[33]  Cecil C. Rousseau,et al.  On ramsey numbers for books , 1978, J. Graph Theory.