Two variants of the size Ramsey number

Andrzej Kurek; Andrzej Ruciński

Discussiones Mathematicae Graph Theory, Tome 25 (2005), p. 141-149 / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let $m (G) = m a x_{F \subseteq G} | E (F) | / | V (F) |$ and define the Ramsey density $m_{i n f} (H, r)$ as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then $m_{i n f} (H, r) = (R - 1) / 2$ , where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál that the size Ramsey number for Kₖ equals $(\binom{R}{2})$ . We also study an on-line version of the size Ramsey number, related to the following two-person game: Painter colors on-line the edges provided by Builder, and Painter’s goal is to avoid a monochromatic copy of Kₖ. The on-line Ramsey number R̅(k;r) is the smallest number of moves (edges) in which Builder can force Painter to lose if r colors are available. We show that R̅(3;2) = 8 and $R ̅ (k; 2) \leq 2 k (\binom{2 k - 2}{k - 1})$ , but leave unanswered the question if R̅(k;2) = o(R²(k;2)).

Publié le : 2005-01-01

Zbl 1074.05062

EUDML-ID : urn:eudml:doc:270499

@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1268,
     author = {Andrzej Kurek and Andrzej Ruci\'nski},
     title = {Two variants of the size Ramsey number},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {25},
     year = {2005},
     pages = {141-149},
     zbl = {1074.05062},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1268}
}

Andrzej Kurek; Andrzej Ruciński. Two variants of the size Ramsey number. Discussiones Mathematicae Graph Theory, Tome 25 (2005) pp. 141-149. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1268/

Bibliographie

[000] [1] N. Alon, C. McDiarmid and B. Reed, Star arboricity, Combinatorica 12 (1992) 375-380, doi: 10.1007/BF01305230.

[001] [2] N. Alon and V. Rödl, Sharp bounds for some multicolor Ramsey numbers, Combinatorica (2005), to appear, doi: 10.1007/s00493-005-0011-9. | Zbl 1090.05049

[002] [3] J. Beck, Achievements games and the probabilistic method, in: Combinatorics, Paul Erdős is Eighty, Bolyai Soc. Math. Stud. 1 (1993) 51-78.

[003] [4] R. Diestel, Graph Theory (Springer, 1996).

[004] [5] P. Erdős, R.J. Faudree, C.C. Rousseau and R.H. Schelp, The size Ramsey number, Period. Math. Hungar. 9 (1978) 145-161, doi: 10.1007/BF02018930. | Zbl 0331.05122

[005] [6] E. Friedgut, V. Rödl, A. Ruciński and P. Tetali, Random graphs with a monochromatic triangle in every edge coloring: a sharp threshold, Memoirs of the AMS, to appear. | Zbl 1087.05052

[006] [7] E. Friedgut, Y. Kohayakawa, V. Rödl, A. Ruciński and P. Tetali, Ramsey games against a one-armed bandit, Combinatorics, Probability and Computing 12 (2003) 515-545, doi: 10.1017/S0963548303005881. | Zbl 1059.05093

[007] [8] R.L. Graham, T. Łuczak, V. Rödl and A. Ruciński, Ramsey properties of families of graphs, J. Combin. Theory (B) 86 (2002) 413-419, doi: 10.1006/jctb.2002.2136. | Zbl 1031.05084

[008] [9] J.A. Grytczuk, M. Hałuszczak and H.A. Kierstead, On-line Ramsey Theory, Electr. J. Combin. (Sep 9, 2004), # R57. | Zbl 1057.05058

[009] [10] E. Györi, B. Rothschild and A. Ruciński, Every graph is contained in a sparsest possible balanced graph, Math. Proc. Cambr. Phil. Soc. 98 (1985) 397-401, doi: 10.1017/S030500410006360X. | Zbl 0582.05048

[010] [11] R.L. Graham, B. Rothschild and J.H. Spencer, Ramsey Theory (2nd ed., Wiley, 1990).

[011] [12] S. Janson, T. Łuczak and A. Ruciński, Random Graphs (Wiley, 2000), doi: 10.1002/9781118032718.

[012] [13] M. Krivelevich, Bounding Ramsey numbers through large deviation inequalities, Random Strutures Algorithms 7 (1995) 145-155, doi: 10.1002/rsa.3240070204. | Zbl 0835.05047

[013] [14] A. Kurek, Arboricity and star arboricity of graphs, in: Proc. of 4th Czechoslovak Symposium on Combinatorics, Graphs and Complexity (1992) 171-173, doi: 10.1016/S0167-5060(08)70624-1. | Zbl 0762.05036

[014] [15] A. Kurek, The density of Ramsey graphs (Ph.D. Thesis, AMU, 1997), in Polish.

[015] [16] A. Kurek and A. Ruciński, Globally sparse vertex-Ramsey graphs, J. Graph Theory 18 (1994) 73-81. | Zbl 0790.05027

[016] [17] R.D. Luce and H. Raiffa, Games and Decisions (Wiley, 1957). | Zbl 0084.15704

[017] [18] T. Łuczak, A. Ruciński and B. Voigt, Ramsey properties of random graphs, J. Combin. Theory (B) 56 (1992) 55-68, doi: 10.1016/0095-8956(92)90006-J. | Zbl 0711.05041

[018] [19] C. Payan, Graphes equilibres at arboricite rationalle, Europ. J. Combin. 7 (1986) 263-270.

[019] [20] S. Radziszowski, Small Ramsey Numbers, Electron. J. Combin. Dynamic Survey DS1 (38pp).

[020] [21] F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 264-286, doi: 10.1112/plms/s2-30.1.264. | Zbl 55.0032.04

[021] [22] V. Rödl and A. Ruciński, Lower bounds on probability thresholds for Ramsey properties, in: Combinatorics, Paul Erdős is Eighty (Vol. 1), Keszthely (Hungary), Bolyai Soc. Math. Studies (1993) 317-346. | Zbl 0790.05078

[022] [23] A. Ruciński, From random graphs to graph theory: Ramsey properties (extended abstract), Graph Theory Notes of New York XX (1991) 8-16.

[023] [24] A. Ruciński, From random graphs to graph theory (survey paper), in: Quo Vadis, Graph Theory?, Annals of Discrete Math. 55 (1993) 265-274.