Hostname: page-component-77c78cf97d-54lbx Total loading time: 0 Render date: 2026-04-24T12:39:21.799Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Bound for the 2/3 Conjecture†

Published online by Cambridge University Press:  23 January 2013

DANIEL KRÁL' ,
CHUN-HUNG LIU ,
JEAN-SÉBASTIEN SERENI ,
PETER WHALEN  and
ZELEALEM B. YILMA
DANIEL KRÁL'
Affiliation:
Institute of Mathematics, DIMAP and Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK (e-mail: D.Kral@warwick.ac.uk)
CHUN-HUNG LIU
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: cliu87@math.gatech.edu, pwhalen3@math.gatech.edu)
JEAN-SÉBASTIEN SERENI
Affiliation:
CNRS (LORIA), Nancy, France (e-mail: sereni@kam.mff.cuni.cz)
PETER WHALEN
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: cliu87@math.gatech.edu, pwhalen3@math.gatech.edu)
ZELEALEM B. YILMA
Affiliation:
LIAFA (Université Denis Diderot), Paris, France (e-mail: Zelealem.Yilma@liafa.jussieu.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours is at least 2n/3. The previous best value, proved by Erdős, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

Keywords

Primary 05C35Secondary 05C55

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 3 , May 2013 , pp. 384 - 393
Copyright
Copyright © Cambridge University Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

†

This work was done in the framework of LEA STRUCO.

References

[1]Baber, R. Turán densities of hypercubes. Submitted. arXiv:1201.3587.Google Scholar
[2]Baber, R. and Talbot, J. (2011) Hypergraphs do jump. Combin. Probab. Comput. 20 161–171.Google Scholar
[3]Baber, R. and Talbot, J. (2012) New Turán densities for 3-graphs. Electron. J. Combin. 19 R22.Google Scholar
[4]Balogh, J., Hu, P., Lidický, B. and Liu, H. Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. Submitted. arXiv:1201.0209.Google Scholar
[5]Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008) Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.CrossRefGoogle Scholar
[6]Cummings, J., Král', D., Pfender, F., Sperfeld, K., Treglown, A. and Young, M. Monochromatic triangles in three-coloured graphs. Submitted. arXiv:1206.1987.Google Scholar
[7]Erdős, P., Faudree, R., Gyárfás, A. and Schelp, R. H. (1989) Domination in colored complete graphs. J. Graph Theory 13 713–718.CrossRefGoogle Scholar
[8]Erdős, P., Faudree, R. J., Gould, R. J., Gyárfás, A., Rousseau, C. and Schelp, R. H. (1990) Monochromatic coverings in colored complete graphs. In Proc. Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing. Congr. Numer. 71 29–38.Google Scholar
[9]Erdős, P. and Hajnal, A. (1989) Ramsey-type theorems. In Combinatorics and Complexity. Discrete Appl. Math. 25 37–52.Google Scholar
[10]Falgas-Ravry, V. and Vaughan, E. R. On applications of Razborov's flag algebra calculus to extremal 3-graph theory. Submitted. arXiv:1110.1623.Google Scholar
[11]Falgas-Ravry, V. and Vaughan, E. R. Turán H-densities for 3-graphs. Submitted. arXiv:1201.4326.Google Scholar
[12]Grzesik, A. (2012) On the maximum number of five-cycles in a triangle-free graph. J. Combin. Theory Ser. B 102 1061–1066.Google Scholar
[13]Hatami, H., Hirst, J. and Norine, S. The inducibility of blow-up graphs. Submitted. arXiv:1108.5699.Google Scholar
[14]Hatami, H., Hladký, J., Král', D., Norine, S. and Razborov, A. On the number of pentagons in triangle-free graphs. J. Combin. Theory Ser. A. Submitted. arXiv:1102.1634.Google Scholar
[15]Hatami, H., Hladký, J., Král', D., Norine, S. and Razborov, A. (2012) Non-three-colourable common graphs exist. Combin. Probab. Comput. 21 734–742.Google Scholar
[16]Hirst, J. The inducibility of graphs on four vertices. Submitted. arXiv:1109.1592.Google Scholar
[17]Hladký, J., Král', D. and Norine, S. Counting flags in triangle-free digraphs. Submitted. arXiv:0908.2791.Google Scholar
[18]Král', D., Liu, C.-H., Sereni, J.-S., Whalen, P. and Yilma, Z. B. A new bound for the 2/3 conjecture. arXiv:1204.2519.Google Scholar
[19]Král', D., Mach, L. and Sereni, J.-S. (2012) A new lower bound based on Gromov's method of selecting heavily covered points. Discrete Comput. Geom. 48 487–498.Google Scholar
[20]Kramer, L., Martin, R. R. and Young, M. On diamond-free subposets of the Boolean lattice. J. Combin. Theory Ser. A. Submitted. arXiv:1205.1501.Google Scholar
[21]Lovász, L. and Szegedy, B. (2006) Limits of dense graph sequences. J. Combin. Theory Ser. B 96 933–957.Google Scholar
[22]Pikhurko, O. Minimum number of k-cliques in graphs with bounded independence number. Submitted. arXiv:1203.4393.Google Scholar
[23]Pikhurko, O. and Razborov, A. Asymptotic structure of graphs with the minimum number of triangles. Submitted. arXiv:1204.2846.Google Scholar
[24]Razborov, A. (2007) Flag algebras. J. Symbolic Logic 72 1239–1282.Google Scholar
[25]Razborov, A. (2008) On the minimal density of triangles in graphs. Combin. Probab. Comput. 17 603–618.Google Scholar
[26]Razborov, A. (2010) On 3-hypergraphs with forbidden 4-vertex configurations. SIAM J. Discrete Math. 24 946–963.Google Scholar
[27]Tychonoff, A. (1930) Über die topologische Erweiterung von Räumen. Math. Ann. 102 544–561.CrossRefGoogle Scholar