Startseite Mathematik New lower bound for the minimal number of edges of simple uniform hypergraph without the property Bk
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

New lower bound for the minimal number of edges of simple uniform hypergraph without the property Bk

  • EMAIL logo
Veröffentlicht/Copyright: 15. Juni 2022
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

A hypergraph H = (V, E) has the property Bk if there exists an assignment of two colors to V such that each edge contains at least k vertices of each color. A hypergraph is called simple if every two edges of it have at most one common vertex. We obtain a new lower bound for the minimal number of edges of n-uniform simple hypergraph without the property Bk.


Originally published in Diskretnaya Matematika (2020) 32, №4, 10–37 (in Russian).


  1. Funding: The reported study was funded by RFBR, project number 19-31-90016.

References

[1] Erdős P., Hajnal A., “On a property of families of sets”, Acta Math. Acad. Sci. Hungar., 12:1 (1961), 87–123.10.1007/BF02066676Suche in Google Scholar

[2] Erdős P., “On a combinatorial problem”, NordiskMat. Tidskr., 11 (1963), 5–10.Suche in Google Scholar

[3] Erdős P., “On a combinatorial problem. II”, Acta Math. Acad. Sci. Hungar., 15:3-4 (1964), 445–447.10.1007/BF01897152Suche in Google Scholar

[4] Kostochka A. V., “Color-critical graphs and hypergraphs with few edges: a survey”, More Sets, Graphs and Numbers, Bolyai Soc. Math. Stud., 15, Springer, Berlin, Heidelberg, 2006,175–197.10.1007/978-3-540-32439-3_9Suche in Google Scholar

[5] Raigorodskii A. M., Shabanov D. A., “The Erdős-Hajnal problem of hypergraph colouring, its generalizations, and related problems”, Russian Math. Surveys, 66:5 (2011), 933–1002.10.1070/RM2011v066n05ABEH004764Suche in Google Scholar

[6] Radhakrishnan J., Srinivasan A., “Improved bounds and algorithms for hypergraph 2-coloring”, Random Struct. Algor., 16:1 (2000), 4–32.10.1002/(SICI)1098-2418(200001)16:1<4::AID-RSA2>3.0.CO;2-2Suche in Google Scholar

[7] Cherkashin D., Kozik J., “A note on random greedy coloring of uniform hypergraphs”, Random Struct. Algor., 47:3 (2015), 407–413.10.1002/rsa.20556Suche in Google Scholar

[8] Shabanov D. A., “On one combinatorial problem of Erdos”, Doklady Mathematics, 69:3 (2004), 359–362.Suche in Google Scholar

[9] Shabanov D. A., “Extremal problems for colourings of uniform hypergraphs”, Izv. Math., 71:6 (2007), 1253–1290.10.1070/IM2007v071n06ABEH002388Suche in Google Scholar

[10] Teplyakov S. M., “Upper bound in the Erdos-Hajnal problem of hypergraph coloring”, Math. Notes, 93:1 (2013), 191–195.10.1134/S0001434613010197Suche in Google Scholar

[11] Shabanov D. A., “Randomized algorithms for colourings of hypergraphs”, Sb. Math., 199:7 (2008), 1089–1110.10.1070/SM2008v199n07ABEH003955Suche in Google Scholar

[12] Rozovskaya A. P., “Combinatorial extremum problems for 2-colorings of hypergraphs”, Math. Notes, 90:4 (2011), 571–583.10.1134/S0001434611090264Suche in Google Scholar

[13] Rozovskaya A.P., “On general two-colorings of uniform hypergraphs”, Dokl. Math., 80:3 (2009), 837–839.10.1134/S1064562409060143Suche in Google Scholar

[14] Demidovich Yu. A., “Some generalizations of the B property problem for an “-uniform hypergraph”, Fundam. iprikl. matem., 23:1 (2020), 95–122 (in Russian).10.1007/s10958-022-05828-6Suche in Google Scholar

[15] Erdős P., Lovász L., “Problems and results on 3-chromatic hypergraphs and some related questions”, Colloq. Math. Soc. Janos. Bolyai, 10 (1973), 609–627.Suche in Google Scholar

[16] Kupavskii A., Shabanov D., “Colourings of uniform hypergraphs with large girth and applications”, Comb., Probab. and Comput., 27:2 (2018), 245–273.10.1017/S0963548317000475Suche in Google Scholar

[17] Szabó Z., “An application of Lovász Local Lemma – a new lower bound for the van der Waerden number”, Random Struct. Algor., 1:3 (1990), 343–360.10.1002/rsa.3240010307Suche in Google Scholar

[18] Kostochka A. V., Kumbhat M., “Coloring uniform hypergraphs with few edges”, Random Struct. Algor., 35:3 (2009), 348–368.10.1002/rsa.20284Suche in Google Scholar

[19] Kozik J., Shabanov D., “Improved algorithms for colorings of simple hypergraphs and applications”, J. Comb. Theory, Ser. B, 116 (2016), 312–332.10.1016/j.jctb.2015.09.004Suche in Google Scholar

[20] Kostochka A.V., Rődl V., “Constructions of sparse uniform hypergraphs with high chromatic number”, Random Struct. Algor., 36:1 (2010), 46–56.10.1002/rsa.20293Suche in Google Scholar

[21] Demidovich Yu. A., Raigorodskii A. M., “2-Colorings of uniform hypergraphs”, Math. Notes, 100:4 (2016), 629–632.10.1134/S0001434616090340Suche in Google Scholar

[22] Demidovich Yu. A., “2-Colorings of hypergraphs with large girth”, Math. Notes, 108:2 (2020), 188–200.10.1134/S0001434620070202Suche in Google Scholar

[23] Semenov A., Shabanov D., “On the weak chromatic number of random hypergraph”, Discr. Appl. Math., 276 (2020), 134–154.10.1016/j.dam.2019.03.025Suche in Google Scholar

[24] Akhmejanova M., Shabanov D., “Coloring hypergraphs with bounded cardinalities of edge intersections”, Discrete Mathematics, 343:4 (2020), 1–11.10.1016/j.disc.2019.111692Suche in Google Scholar

[25] Semenov A. S., “Two-colorings of a random hypergraph”, Theory Probab. Appl., 64:1 (2019), 59–77.10.1137/S0040585X97T989398Suche in Google Scholar

[26] Cherkashin D., Petrov F., “On small “-uniform hypergraphs with positive discrepancy”, J. Comb. Theory, Ser. B, 139 (2019), 353–359.10.1016/j.jctb.2019.04.001Suche in Google Scholar

[27] Ayre P., Coja-Oghlan A., Greenhill C., “Hypergraph coloring up to condensation”, Random Struct. Algor., 54:4 (2019), 615–652.10.1002/rsa.20824Suche in Google Scholar

[28] Duraj L., Gutowski G., Kozik J., “A note on two-colorability of nonuniform hypergraphs”, 45th Int. Colloq. Automata, Languages, and Programm. (ICALP 2018), Leibniz Int. Proc. in Informatics, 107, Schloss Dagstuhl – Leibniz-Zentrum für Informatik http://drops.dagstuhl.de/opus/volltexte/2018/9050, 2018, 46:1–46:13.Suche in Google Scholar

[29] Dyer M., Frieze A., Greenhill C., “On the chromatic number of a random hypergraph”, J. Comb. Theory, Ser. B, 113 (2015), 68–122.10.1016/j.jctb.2015.01.002Suche in Google Scholar

[30] Balogh J., Cherkashin D., Kiselev S., “Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs”, Europ. J. Combin., 79 (2019), 228–236.10.1016/j.ejc.2019.03.004Suche in Google Scholar

[31] Raigorodskii A. M., Cherkashin D. D., “Extremal problems in hypergraph colourings”, Russian Math. Surveys, 75:1 (2020), 89–146.10.1070/RM9905Suche in Google Scholar

Received: 2020-11-24
Published Online: 2022-06-15
Published in Print: 2022-06-27

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 25.3.2026 von https://www.degruyterbrill.com/document/doi/10.1515/dma-2022-0015/html
Button zum nach oben scrollen