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Short Proofs of Some Extremal Results

  • DAVID CONLON (a1), JACOB FOX (a2) and BENNY SUDAKOV (a3)

Abstract

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

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[1]Akiyama, J. and Watanabe, M. (1987) Maximum induced forests of planar graphs. Graphs Combin. 3 201202.
[2]Albertson, M. and Haas, R. (1998) A problem raised at the DIMACS Graph Coloring Week.
[3]Alon, N. (2003) Problems and results in extremal combinatorics I. Discrete Math. 273 3153.
[4]Alon, N. (2008) Problems and results in extremal combinatorics II. Discrete Math. 308 44604472.
[5]Alon, N. and Krivelevich, M. (1997) Constructive bounds for a Ramsey-type problem. Graphs Combin. 13 217225.
[6]Alon, N., Mubayi, D. and Thomas, R. (2001) Large induced forests in sparse graphs. J. Graph Theory 38 113123.
[7]Balister, P., Lehel, J. and Schelp, R. H. (2006) Ramsey unsaturated and saturated graphs. J. Graph Theory 51 2232.
[8]Beck, J. (1993) Achievement games and the probabilistic method. In Combinatorics: Paul Erdős is Eighty, Vol. 1, Bolyai Mathematical Society, pp. 5178.
[9]Brown, T. C., Chung, F. R. K., Erdős, P. and Graham, R. L. (1985) Quantitative forms of a theorem of Hilbert. J. Combin. Theory Ser. A 38 210216.
[10]Butterfield, J., Grauman, T., Kinnersley, W. B., Milans, K. G., Stocker, C. and West, D. B. (2011) On-line Ramsey theory for bounded degree graphs. Electron. J. Combin. 18 #136.
[11]Cavers, M. and Verstraëte, J. (2008) Clique partitions of complements of forests and bounded degree graphs. Discrete Math. 308 20112017.
[12]Chang, Y. (1996) A bound for Wilson's theorem III. J. Combin. Des. 4 8393.
[13]Conlon, D. (2009) Hypergraph packing and sparse bipartite Ramsey numbers. Combin. Probab. Comput. 18 913923.
[14]Conlon, D. (2009/10) On-line Ramsey numbers. SIAM J. Discrete Math. 23 19541963.
[15]Conlon, D. (2013) The Ramsey number of dense graphs. Bull. London Math. Soc. 45 483496.
[16]Conlon, D., Fox, J. and Sudakov, B. (2010) Hypergraph Ramsey numbers. J. Amer. Math. Soc. 23 247266.
[17]Dudek, A. and Mubayi, D. On generalized Ramsey numbers for 3-uniform hypergraphs. J. Graph Theory. Preprint.
[18]Dudek, A., Retter, T. and Rödl, V. On generalized Ramsey numbers of Erdős and Rogers. Preprint.
[19]Dudek, A. and Rödl, V. (2011) On Ks-free subgraphs in Ks+k-free graphs and vertex Folkman numbers. Combinatorica 31 3953.
[20]Erdős, P. and Gallai, T. (1961) On the minimal number of vertices representing the edges of a graph. Publ. Math. Inst. Hungar. Acad. Sci. 6 181202.
[21]Erdős, P. and Rado, R. (1952) Combinatorial theorems on classifications of subsets of a given set. Proc. London Math. Soc. 3 417439.
[22]Erdős, P. and Rogers, C. A. (1962) The construction of certain graphs. Canad. J. Math. 14 702707.
[23]Fox, J. and Sudakov, B. (2009) Density theorems for bipartite graphs and related Ramsey-type results. Combinatorica 29 153196.
[24]Friedgut, E., Kohayakawa, Y., Rödl, V., Ruciński, A. and Tetali, P. (2003) Ramsey games against a one-armed bandit. Combin. Probab. Comput. 12 515545.
[25]Gregory, D. A., McGuinness, S. and Wallis, W. (1986) Clique partitions of the cocktail party graph. Discrete Math. 59 267273.
[26]Grytczuk, J. A., Hałuszczak, M. and Kierstead, H. A. (2004) On-line Ramsey theory. Electron. J. Combin. 11 #57.
[27]Gunderson, D. S., Rödl, V. and Sidorenko, A. (1999) Extremal problems for sets forming Boolean algebras and complete partite hypergraphs. J. Combin. Theory Ser. A 88 342367.
[28]Hegyvári, N. (1996) On representation problems in the additive number theory. Acta Math. Hungar. 72 3544.
[29]Hegyvári, N. (1999) On the dimension of the Hilbert cubes. J. Number Theory 77 326330.
[30]Hilbert, D. (1892) Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten. J. Reine Angew. Math. 110 104129.
[31]Kierstead, H. A. and Konjevod, G. (2009) Coloring number and on-line Ramsey theory for graphs and hypergraphs. Combinatorica 29 4964.
[32]Kowalik, Ł, Lužar, B. and Škrekovski, R. (2010) An improved bound on the largest induced forests for triangle-free planar graphs. Discrete Math. Theor. Comput. Sci. 12 87100.
[33]Krivelevich, M. (1994) Ks-free graphs without large Kr-free subgraphs. Combin. Probab. Comput. 3 349354.
[34]Krivelevich, M. (1995) Bounding Ramsey numbers through large deviation inequalities. Random Struct. Alg. 7 145155.
[35]Kurek, A. and Ruciński, A. (2005) Two variants of the size Ramsey number. Discuss. Math. Graph Theory 25 141149.
[36]Marciniszyn, M., Spöhel, R. and Steger, A. (2009) Online Ramsey games in random graphs. Combin. Probab. Comput. 18 271300.
[37]Marciniszyn, M., Spöhel, R. and Steger, A. (2009) Upper bounds for online Ramsey games in random graphs. Combin. Probab. Comput. 18 259270.
[38]Nguyen, H. and Vu, V. (2011) Optimal inverse Littlewood–Offord theorems. Adv. Math. 226 52985319.
[39]Orlin, J. (1977) Contentment in graph theory. Indag. Math. 39 406424.
[40]Ramsey, F. P. (1930) On a problem of formal logic. Proc. London Math. Soc. Ser. 2 30 264286.
[41]Salavatipour, M. R. (2006) Large induced forests in triangle-free planar graphs. Graphs Combin. 22 113126.
[42]Shearer, J. B. (1995) On the independence number of sparse graphs. Random Struct. Alg. 7 269271.
[43]Skokan, J. and Stein, M. Cycles are strongly Ramsey-unsaturated. Combin. Probab. Comput., to appear.
[44]Sudakov, B. (2005) A new lower bound for a Ramsey-type problem. Combinatorica 25 487498.
[45]Sudakov, B. (2005) Large Kr-free subgraphs in Ks-free graphs and some other Ramsey-type problems. Random Struct. Alg. 26 253265.
[46]Sudakov, B. (2011) A conjecture of Erdős on graph Ramsey numbers. Adv. Math. 227 601609.
[47]Szemerédi, E. (1969) On sets of integers containing no four elements in arithmetic progression. Acta Math. Acad. Sci. Hungar. 20 199245.
[48]Tao, T. and Vu, V. (2009) Inverse Littlewood–Offord theorems and the condition number of random discrete matrices. Ann. of Math. 169 595632.
[49]Tao, T. and Vu, V. (2010) A sharp inverse Littlewood–Offord theorem. Random Struct. Alg. 37 525539.
[50]Wallis, W. D. (1985) Clique partitions of the complement of a one-factor. Congr. Numer. 46 317319.
[51]Wallis, W. D. (1990) Finite planes and clique partitions. Contemp. Math. 111 279285.
[52]Wilson, R. M. (1974) Constructions and uses of pairwise balanced designs. In Combinatorics: Proc. NATO Advanced Study Inst., Breukelen, 1974, Part 1: Theory of Designs, Finite Geometry and Coding Theory, No. 55 of Math. Centre Tracts, Math Centrum, Amsterdam, pp. 1841.
[53]Wolfovitz, G.K 4-free graphs without large induced triangle-free subgraphs. Combinatorica, to appear.

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