Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-npccv Total loading time: 0.377 Render date: 2022-09-30T17:37:16.565Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Minimum Number of Monotone Subsequences of Length 4 in Permutations

Published online by Cambridge University Press:  19 December 2014

JÓZSEF BALOGH
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: jobal@math.uiuc.edu, pinghu1@math.uiuc.edu, lidicky@illinois.edu) Bolyai Institute, University of Szeged, Szeged, Hungary (e-mail: udvarib@math.u-szeged.hu)
PING HU
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: jobal@math.uiuc.edu, pinghu1@math.uiuc.edu, lidicky@illinois.edu)
BERNARD LIDICKÝ
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: jobal@math.uiuc.edu, pinghu1@math.uiuc.edu, lidicky@illinois.edu) Charles University, Prague, Czech Republic
OLEG PIKHURKO
Affiliation:
Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK (e-mail: O.Pikhurko@warwick.ac.uk, honza@ucw.cz)
BALÁZS UDVARI
Affiliation:
Bolyai Institute, University of Szeged, Szeged, Hungary (e-mail: udvarib@math.u-szeged.hu)
JAN VOLEC
Affiliation:
Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK (e-mail: O.Pikhurko@warwick.ac.uk, honza@ucw.cz)

Abstract

We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least

\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}
Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromatic K4 is minimized. We show that all the extremal colourings must contain monochromatic K4 only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

Type
Paper
Copyright
Copyright © Cambridge University Press 2014 

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.)

References

[1]Albert, M. H., Atkinson, M. D., Handley, C. C., Holton, D. A. and Stromquist, W. (2002) On packing densities of permutations. Electron. J. Combin. 9 R5.Google Scholar
[2]Alon, N. and Shapira, A. (2008) A characterization of the (natural) graph properties testable with one-sided error. SIAM J. Comput. 37 17031727.CrossRefGoogle Scholar
[3]Baber, R. Turán densities of hypercubes. Submitted.Google Scholar
[4]Baber, R. (2011) Some results in extremal combinatorics. Dissertation.Google Scholar
[5]Baber, R. and Talbot, J. (2011) Hypergraphs do jump. Combin. Probab. Comput. 20 161171.CrossRefGoogle Scholar
[6]Baber, R. and Talbot, J. (2014) A solution to the 2/3 conjecture. SIAM J. Discrete Math. 28 756766.CrossRefGoogle Scholar
[7]Balogh, J., Hu, P., Lidický, B. and Liu, H. (2014) Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. European J. Combin. 35 7585.CrossRefGoogle Scholar
[8]Borchers, B. (1999) CSDP: A C library for semidefinite programming. Optimization Methods and Software 11 613623.CrossRefGoogle Scholar
[9]Cummings, J., Kráľ, D., Pfender, F., Sperfeld, K., Treglown, A. and Young, M. (2013) Monochromatic triangles in three-coloured graphs. J. Combin. Theory Ser. B 103 489503.CrossRefGoogle Scholar
[10]Das, S., Huang, H., Ma, J., Naves, H. and Sudakov, B. (2013) A problem of Erdős on the minimum number of k-cliques. J. Combin. Theory Ser. B 103 344373.CrossRefGoogle Scholar
[11]Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Math. 2 463470.Google Scholar
[12]Falgas-Ravry, V. and Vaughan, E. R. (2013) Applications of the semi-definite method to the Turán density problem for 3-graphs. Combin. Probab. Comput. 22 2154.CrossRefGoogle Scholar
[13]Giraud, G. (1979) Sur le problème de Goodman pour les quadrangles et la majoration des nombres de Ramsey. J. Combin. Theory Ser. B 27 237253.CrossRefGoogle Scholar
[14]Glebov, R., Kráľ, D. and Volec, J. (2013) A problem of Erdős and Sós on 3-graphs. In The Seventh European Conference on Combinatorics, Graph Theory and Applications, Vol. 16 of CRM Series, Ed. Norm., Pisa. pp. 38.CrossRefGoogle Scholar
[15]Goodman, A. W. (1959) On sets of acquaintances and strangers at any party. Amer. Math. Monthly 66 778783.CrossRefGoogle Scholar
[16]Grzesik, A. (2012) On the maximum number of five-cycles in a triangle-free graph. J. Combin. Theory Ser. B 102 10611066.CrossRefGoogle Scholar
[17]Hatami, H., Hladký, J., Kráľ, D., Norine, S. and Razborov, A. (2012) Non-three-colourable common graphs exist. Combin. Probab. Comput. 21 734742.CrossRefGoogle Scholar
[18]Hatami, H., Hladký, J., Kráľ, D., Norine, S. and Razborov, A. (2013) On the number of pentagons in triangle-free graphs. J. Combin. Theory Ser. A 120 722732.CrossRefGoogle Scholar
[19]Hladký, J., Kráľ, D. and Norine, S. (2009) Counting flags in triangle-free digraphs. In European Conference on Combinatorics, Graph Theory and Applications: EuroComb 2009, Vol. 34 of Electron. Notes Discrete Math., Elsevier, pp. 621625.Google Scholar
[20]Kráľ, D., Liu, C.-H., Sereni, J.-S., Whalen, P. and Yilma, Z. B. (2013) A new bound for the 2/3 conjecture. Combin. Probab. Comput. 22 384393.CrossRefGoogle Scholar
[21]Kráľ, 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 487498.CrossRefGoogle Scholar
[22]Myers, J. S. (2002/03) The minimum number of monotone subsequences. Electron. J. Combin. 9 R4.Google Scholar
[23]Nieß, S. Counting monochromatic copies of K 4: A new lower bound for the Ramsey multiplicity problem. Submitted.Google Scholar
[24]Pikhurko, O. (2011) The minimum size of 3-graphs without a 4-set spanning no or exactly three edges. European J. Combin. 32 11421155.CrossRefGoogle Scholar
[25]Pikhurko, O. and Razborov, A. Asymptotic structure of graphs with the minimum number of triangles. arXiv:1204.2846. Submitted.Google Scholar
[26]Pikhurko, O. and Vaughan, E. R. (2013) Minimum number of k-cliques in graphs with bounded independence number. Combin. Probab. Comput. 22 910934.CrossRefGoogle Scholar
[27]Presutti, C. B. (2008) Determining lower bounds for packing densities of non-layered patterns using weighted templates. Electron. J. Combin. 15 R50.Google Scholar
[28]Presutti, C. B. and Stromquist, W. (2010) Packing rates of measures and a conjecture for the packing density of 2413. In Permutation Patterns, Vol. 376 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 287316.CrossRefGoogle Scholar
[29]Razborov, A. A. (2007) Flag algebras. J. Symbolic Logic 72 12391282.CrossRefGoogle Scholar
[30]Razborov, A. A. (2013) Flag algebras: An interim report. In The Mathematics of Paul Erdős II, Springer, pp. 207232.CrossRefGoogle Scholar
[31]Razborov, A. A. (2013) On the Caccetta–Häggkvist conjecture with forbidden subgraphs. J. Graph Theory 74 236248.CrossRefGoogle Scholar
[32]Samotij, W. and Sudakov, B. On the number of monotone sequences. Submitted.Google Scholar
[33]Sperfeld, K. The inducibility of small oriented graphs. Submitted.Google Scholar
[34]Sperfeld, K. (2011) On the minimal monochromatic K 4-density.Google Scholar
[35]Stein, W.et al. (2013) Sage Mathematics Software: Version 5.6. The SAGE Development Team. http://www.sagemath.orgGoogle Scholar
[36]Thomason, A. (1989) A disproof of a conjecture of Erdős in Ramsey theory. J. London Math. Soc. (2) 39 246255.CrossRefGoogle Scholar
[37]Vaughan, E. R. (2013) Flagmatic: Version 2.0. http://www.flagmatic.orgGoogle Scholar
[38]Yamashita, M., Fujisawa, K., Nakata, K., Nakata, M., Fukuda, M., Kobayashi, K. and Goto, K. (2010) A high-performance software package for semidefinite programs: SDPA 7. Research Report B-460 Dept. of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, Japan, September, 2010.Google Scholar
15
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Minimum Number of Monotone Subsequences of Length 4 in Permutations
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Minimum Number of Monotone Subsequences of Length 4 in Permutations
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Minimum Number of Monotone Subsequences of Length 4 in Permutations
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *