Conlon, David (2019) The Ramsey number of books. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20190819-170924800
![]() |
PDF
- Submitted Version
See Usage Policy. 161kB |
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20190819-170924800
Abstract
We show that in every two-colouring of the edges of the complete graph K_N there is a monochromatic K_k which can be extended in at least (1 + o_(k)(1))2^(-k)N ways to a monochromatic K_(k+1). This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book B_n^(k) to be the graph consisting of n copies of K_(k+1) all sharing a common K_k, we show that the Ramsey number r(B_n^(k)) = 2^(k)n + o_(k)(n). In this form, our result answers a question of Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.
Item Type: | Report or Paper (Discussion Paper) | ||||||||
---|---|---|---|---|---|---|---|---|---|
Related URLs: |
| ||||||||
ORCID: |
| ||||||||
Additional Information: | Research supported by a Royal Society University Research Fellowship and ERC Starting Grant 676632. This paper was partially written while I was visiting the California Institute of Technology as a Moore Distinguished Scholar and I am extremely grateful for their kind support. I am also much indebted to Jacob Fox, Lisa Sauermann and Yuval Wigderson for pointing out a subtle error in the first version of this paper. Finally, I would like to thank the anonymous reviewers for several helpful remarks which improved the presentation. | ||||||||
Funders: |
| ||||||||
Record Number: | CaltechAUTHORS:20190819-170924800 | ||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20190819-170924800 | ||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||
ID Code: | 98030 | ||||||||
Collection: | CaltechAUTHORS | ||||||||
Deposited By: | Melissa Ray | ||||||||
Deposited On: | 20 Aug 2019 23:12 | ||||||||
Last Modified: | 03 Oct 2019 21:37 |
Repository Staff Only: item control page