Skip to main content

Ramsey Numbers of Books and Quasirandomness


The book graph \(B_n^{(k)}\) consists of n copies of Kk+1 joined along a common Kk. The Ramsey numbers of \(B_n^{(k)}\) are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed k, thus answering an old question of Erdős, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemerédi’s regularity lemma, thus providing much tighter control on the error term. Finally, we prove a conjecture of Nikiforov, Rousseau, and Schelp by showing that all extremal colorings for this Ramsey problem are quasirandom.

This is a preview of subscription content, access via your institution.


  1. F. R. K. Chung, R. L. Graham and R. M. Wilson: Quasi-random graphs, Combinatorica 9 (1989), 345–362.

    MathSciNet  Article  Google Scholar 

  2. D. Conlon: A new upper bound for diagonal Ramsey numbers, Ann. of Math. 170 (2009), 941–960.

    MathSciNet  Article  Google Scholar 

  3. D. Conlon: The Ramsey number of books, Adv. Combin., Paper No. 3, 2019.

  4. D. Conlon and J. Fox: Bounds for graph regularity and removal lemmas, Geom. Funct. Anal. 22 (2012), 1191–1256.

    MathSciNet  Article  Google Scholar 

  5. R. A. Duke, H. Lefmann and V. Rödl: A fast approximation algorithm for computing the frequencies of subgraphs in a given graph, SIAM J. Comput. 24 (1995), 598–620.

    MathSciNet  Article  Google Scholar 

  6. P. Erdős: On the number of complete subgraphs contained in certain graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962), 459–464.

    MathSciNet  MATH  Google Scholar 

  7. P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp: The size Ramsey number, Period. Math. Hungar. 9 (1978), 145–161.

    MathSciNet  Article  Google Scholar 

  8. P. Erdős and A. Szemerédi: On a Ramsey type theorem, Period. Math. Hungar. 2 (1972), 295–299.

    MathSciNet  Article  Google Scholar 

  9. J. Fox and B. Sudakov: Induced Ramsey-type theorems, Adv. Math. 219 (2008), 1771–1800.

    MathSciNet  Article  Google Scholar 

  10. A. Frieze and R. Kannan: The regularity lemma and approximation schemes for dense problems, in: 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), 12–20, IEEE Comput. Soc. Press, Los Alamitos, CA, 1996.

    Google Scholar 

  11. A. Frieze and R. Kannan: Quick approximation to matrices and applications, Combinatorica 19 (1999), 175–220.

    MathSciNet  Article  Google Scholar 

  12. A. W. Goodman: On sets of acquaintances and strangers at any party, Amer. Math. Monthly 66 (1959), 778–783.

    MathSciNet  Article  Google Scholar 

  13. M. Jenssen, P. Keevash, E. Long and L. Yepremyan: Distinct degrees in induced subgraphs, Proc. Amer. Math. Soc. 148 (2020), 3835–3846.

    MathSciNet  Article  Google Scholar 

  14. J. Komlós, A. Shokoufandeh, M. Simonovits And E. Szemerédi: The regularity lemma and its applications in graph theory, in: Theoretical aspects of computer science (Tehran, 2000), volume 2292 of Lecture Notes in Comput. Sci., 84–112, Springer, Berlin, 2002.

    Google Scholar 

  15. M. Kwan and B. Sudakov: Proof of a conjecture on induced subgraphs of Ramsey graphs, Trans. Amer. Math. Soc. 372 (2019), 5571–5594.

    MathSciNet  Article  Google Scholar 

  16. L. Lovász: Large networks and graph limits, volume 60 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2012.

    Google Scholar 

  17. V. Nikiforov, C. C. Rousseau and R. H. Schelp: Book Ramsey numbers and quasi-randomness, Combin. Probab. Comput. 14 (2005), 851–860.

    MathSciNet  Article  Google Scholar 

  18. H. J. Prömel and V. Rödl: Non-Ramsey graphs are c log n-universal, J. Combin. Theory Ser. A 88 (1999), 379–384.

    MathSciNet  Article  Google Scholar 

  19. F. P. Ramsey: On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1929), 264–286.

    MathSciNet  MATH  Google Scholar 

  20. C. C. Rousseau and J. Sheehan: On Ramsey numbers for books, J. Graph Theory 2 (1978), 77–87.

    MathSciNet  Article  Google Scholar 

  21. V. T. Sós: Induced subgraphs and Ramsey colorings, Presented at the 16th International Conference on Random Structures and Algorithms, 2013.

  22. J. M. Steele: The Cauchy-Schwarz master class: An introduction to the art of mathematical inequalities, MAA Problem Books Series, Mathematical Association of America, Washington, DC, Cambridge University Press, Cambridge, 2004.

    Book  Google Scholar 

  23. A. Thomason: On finite Ramsey numbers, European J. Combin. 3 (1982), 263–273.

    MathSciNet  Article  Google Scholar 

  24. A. Thomason: Pseudorandom graphs, in: Random graphs’ 85 (Poznań, 1985), volume 144 of North-Holland Math. Stud., 307–331, North-Holland, Amsterdam, 1987.

    Google Scholar 

  25. A. Thomason: A disproof of a conjecture of Erdős in Ramsey theory, J. London Math. Soc. 39 (1989), 246–255.

    MathSciNet  Article  Google Scholar 

  26. Y. Zhao: Lecture notes on graph theory and additive combinatorics,, 2019.

Download references


We would like to thank Freddie Illingworth for pointing out an error in an earlier draft of this paper. We would also like to thank the anonymous referees for their careful reviews and helpful suggestions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to David Conlon.

Additional information

Research supported by ERC Starting Grant 676632 and NSF Award DMS-2054452.

Research supported by a Packard Fellowship and by NSF Career Award DMS-1352121.

Research supported by NSF GRFP Grant DGE-1656518.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Conlon, D., Fox, J. & Wigderson, Y. Ramsey Numbers of Books and Quasirandomness. Combinatorica 42, 309–363 (2022).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification (2010)

  • 05C55
  • 05D10