Skip to main content Accessibility help
×
Home

Ramsey properties of randomly perturbed graphs: cliques and cycles

  • Shagnik Das (a1) and Andrew Treglown (a2)

Abstract

Given graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem.

We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4).

To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

Copyright

Corresponding author

*Corresponding author. Email: a.c.treglown@bham.ac.uk

Footnotes

Hide All

Research supported by GIF grant G-1347-304.6/2016.

Research supported by EPSRC grant EP/M016641/1.

Footnotes

References

Hide All
[1]Aigner-Horev, E. and Person, Y. (2019) Monochromatic Schur triples in randomly perturbed dense sets of integers. SIAM J. Discrete Math. 33 21752180.
[2]Alon, N. and Spencer, J. (2015) The Probabilistic Method, Wiley.
[3]Balogh, J., Morris, R. and Samotij, W. (2015) Independent sets in hypergraphs. J. Amer. Math. Soc. 28 669709.
[4]Balogh, J., Treglown, A. and Wagner, A. Z. (2019) Tilings in randomly perturbed dense graphs. Combin. Probab. Comput. 28 159176.
[5]Bedenknecht, W., Han, J., Kohayakawa, Y. and Mota, G. O. (2019) Powers of tight Hamilton cycles in randomly perturbed hypergraphs. Random Struct. Algorithms 55 795807.
[6]Bennett, P., Dudek, A. and Frieze, A. (2017) Adding random edges to create the square of a Hamilton cycle. arXiv:1710.02716
[7]Bohman, T., Frieze, A., Krivelevich, M. and Martin, R. (2004) Adding random edges to dense graphs. Random Struct. Algorithms 24 105117.
[8]Bohman, T., Frieze, A. and Martin, R. (2003) How many edges make a dense graph Hamiltonian? Random Struct. Algorithms 22 3342.
[9]Böttcher, J., Han, J., Kohayakawa, Y., Montgomery, R., Parczyk, O. and Person, Y. (2019) Universality for bounded degree spanning trees in randomly perturbed graphs. Random Struct. Algorithms 55 854864.
[10]Böttcher, J., Montgomery, R., Parczyk, O. and Person, Y. (2020) Embedding spanning bounded degree graphs in randomly perturbed graphs. Mathematika 66 422447.
[11]Chung, F. (1997) Open problems of Paul Erdős in graph theory. J. Graph Theory 25 336.
[12]Conlon, D. (2009) A new upper bound for diagonal Ramsey numbers. Ann. of Math. 170 941960.
[13]Conlon, D. and Gowers, W. T. (2016) Combinatorial theorems in sparse random sets. Ann. Math. 84 367454.
[14]Day, A. N. and Johnson, J. R. (2017) Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B 124 5663.
[15]Dudek, A., Reiher, C., Ruciński, A. and Schacht, M. (2020) Powers of Hamiltonian cycles in randomly augmented graphs. Random Struct. Algorithms 56 122141.
[16]Erdős, P. and Graham, R. L. (1973) On partition theorems for finite graphs. Colloq. Math. Soc. János Bolyai. 10 515527.
[17]Erdős, P. and Stone, A. H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52 10871091.
[18]Fox, J. and Sudakov, B. (2011) Dependent random choice. Random Struct. Algorithms 38 6899.
[19]Friedgut, E., Rödl, V. and Schacht, M. (2010) Ramsey properties of random discrete structures. Random Struct. Algorithms 37 407436.
[20]Gugelmann, L., Nenadov, R., Person, Y., Škorić, N., Steger, A. and Thomas, H. (2017) Symmetric and asymmetric Ramsey properties in random hypergraphs. Forum Math. Sigma 5 E28.
[21]Han, J. and Zhao, Y. Hamiltonicity in randomly perturbed hypergraphs. J. Combin. Theory Ser. B, to appear.
[22]Hancock, R., Staden, K. and Treglown, A. (2019) Independent sets in hypergraphs and Ramsey properties of graphs and the integers. SIAM J. Discrete Math. 33 153188.
[23]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.
[24]Joos, F. and Kim, J. (2020) Spanning trees in randomly perturbed graphs. Random Struct. Algorithms 56 169219.
[25]Kohayakawa, Y. and Kreuter, B. (1997) Threshold functions for asymmetric Ramsey properties involving cycles. Random Struct. Algorithms 11 245276.
[26]Komlós, J. and Simonovits, M. (1996) Szemerédi’s Regularity Lemma and its applications in graph theory. In Combinatorics: Paul Erdös is Eighty, Vol. 2 (Miklós, D., Sós, V. T. and Szőnyi, T. eds.), pp. 295352, János Bolyai Mathematical Society.
[27]Kreuter, B. (1996) Threshold functions for asymmetric Ramsey properties with respect to vertex colorings. Random Struct. Algorithms 9 335348.
[28]Krivelevich, M., Kwan, M. and Sudakov, B. (2016) Cycles and matchings in randomly perturbed digraphs and hypergraphs. Combin. Probab. Comput. 25 909927.
[29]Krivelevich, M., Kwan, M. and Sudakov, B. (2017) Bounded-degree spanning trees in randomly perturbed graphs. SIAM J. Discrete Math. 31 155171.
[30]Krivelevich, M., Sudakov, B. and Tetali, P. (2006) On smoothed analysis in dense graphs and formulas. Random Struct. Algorithms 29 180193.
[31]łuczak, T., Ruciński, A. and Voigt, B. (1992) Ramsey properties of random graphs. J. Combin. Theory Ser. B 56 5568.
[32]Marciniszyn, M., Skokan, J., Spöhel, R. and Steger, A. (2009) Asymmetric Ramsey properties of random graphs involving cliques. Random Struct. Algorithms 34 419453.
[33]McDowell, A. and Mycroft, R. (2018) Hamilton l-cycles in randomly perturbed hypergraphs. Electron. J. Combin. 25 P4.36.
[34]Mousset, F., Nenadov, R. and Samotij, W. (2018) Towards the Kohayakawa–Kreuter conjecture on asymmetric Ramsey properties. Comb. Probab. Comput. arXiv:1808.05070
[35]Nenadov, R. and Steger, A. (2016) A short proof of the random Ramsey theorem. Combin. Probab. Comput. 25 130144.
[36]Nenadov, R. and Trujić, M. (2018) Sprinkling a few random edges doubles the power. arXiv:1811.09209
[37]Powierski, E. (2019) Ramsey properties of randomly perturbed dense graphs. arXiv:1902.02197
[38]Rödl, V. and Ruciński, A. (1993) Lower bounds on probability thresholds for Ramsey properties. In Combinatorics: Paul Erdős is Eighty, Vol. 1 (Miklós, D., Sós, V. T. and Szőnyi, T., eds), pp. 317346, János Bolyai Mathematical Society.
[39]Rödl, V. and Ruciński, A. (1994) Random graphs with monochromatic triangles in every edge coloring. Random Struct. Algorithms 5 253270.
[40]Rödl, V. and Ruciński, A. (1995) Threshold functions for Ramsey properties. J. Amer. Math. Soc. 8 917942.
[41]Saxton, D. and Thomason, A. (2015) Hypergraph containers. Inventio Math. 201 925992.
[42]Spencer, J. H. (1975) Ramsey’s theorem: a new lower bound. J. Combin. Theory Ser. A 18 108115.
[43]Szemerédi, E. (1976) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes, Vol. 260 of Colloques Internationaux CNRS, pp. 399401.
[44]Turán, P. (1941) On an extremal problem in graph theory (in Hungarian). Math. Fiz. Lapok 48 436452.

Ramsey properties of randomly perturbed graphs: cliques and cycles

  • Shagnik Das (a1) and Andrew Treglown (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.