Book contents
- Frontmatter
- Contents
- Preface
- 1 Ramsey classes: examples and constructions
- 2 Recent developments in graph Ramsey theory
- 3 Controllability and matchings in random bipartite graphs
- 4 Some old and new problems in combinatorial geometry I: around Borsuk's problem
- 5 Randomly generated groups
- 6 Curves over finite fields and linear recurring sequences
- 7 New tools and results in graph minor structure theory
- 8 Well quasi-order in combinatorics: embeddings and homomorphisms
- 9 Constructions of block codes from algebraic curves over finite fields
- References
2 - Recent developments in graph Ramsey theory
Published online by Cambridge University Press: 05 July 2015
Book contents
- Frontmatter
- Contents
- Preface
- 1 Ramsey classes: examples and constructions
- 2 Recent developments in graph Ramsey theory
- 3 Controllability and matchings in random bipartite graphs
- 4 Some old and new problems in combinatorial geometry I: around Borsuk's problem
- 5 Randomly generated groups
- 6 Curves over finite fields and linear recurring sequences
- 7 New tools and results in graph minor structure theory
- 8 Well quasi-order in combinatorics: embeddings and homomorphisms
- 9 Constructions of block codes from algebraic curves over finite fields
- References
Summary
Abstract
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of KN contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress.
1 Introduction
In its broadest sense, the term Ramsey theory refers to any mathematical statement which says that a structure of a given kind is guaranteed to contain a large well-organised substructure. There are examples of such statements in many areas, including geometry, number theory, logic and analysis. For example, a key ingredient in the proof of the Bolzano–Weierstrass theorem in real analysis is a lemma showing that any infinite sequence must contain an infinite monotone subsequence.
A classic example from number theory, proved by van der Waerden [212] in 1927, says that if the natural numbers are coloured in any fixed number of colours then one of the colour classes contains arbitrarily long arithmetic progressions. This result has many generalisations. The most famous, due to Szemerédi [206], says that any subset of the natural numbers of positive upper density contains arbitrarily long arithmetic progressions. This result has many generalisations. The most famous, due to Szemerédi [206], says that any subset of the natural numbers of positive upper density contains arbitrarily long arithmetic progressions.
- Type
- Chapter
- Information
- Surveys in Combinatorics 2015 , pp. 49 - 118Publisher: Cambridge University PressPrint publication year: 2015
References
[3] , , and , Sizes of induced subgraphs of Ramsey graphs, Combin. Probab. Comput. 18 (2009), 459–476.Google Scholar
[4] and , Constructive bounds for a Ramsey-type problem, Graphs Combin. 13 (1997), 217–225.Google Scholar
[5] , and , Turáan numbers of bipartite graphs and related Ramsey-type questions, Combin. Probab. Comput. 12 (2003), 477–494.Google Scholar
[6] , and , Large nearly regular induced subgraphs, SIAM J. Discrete Math. 22 (2008), 1325–1337.Google Scholar
[7] , , , and , Crossing patterns of semi-algebraic sets, J. Combin. Theory Ser. A 111 (2005), 310–326.Google Scholar
[8] , and , Ramsey-type theorems with forbidden subgraphs, Combinatorica 21 (2001), 155–170.Google Scholar
[9] and , Sharp bounds for some multicolor Ramsey numbers, Combinatorica 25 (2005), 125–141.Google Scholar
[10] , and , A separator theorem for nonplanar graphs, J. Amer. Math. Soc. 3 (1990), 801–808.Google Scholar
[11] and , The Probabilistic Method, 3rd edition, Wiley, 2007.
[12] , , and , Ramsey numbers of ordered graphs, preprint.
[13] , and , Independent sets in hypergraphs, to appear in J. Amer. Math. Soc.
[14] , , and , 2-source dispersers for no(1) entropy, and Ramsey graphs beating the Frankl–Wilson construction, Ann. of Math. 176 (2012), 1483–1543.Google Scholar
[15] , An upper bound for diagonal Ramsey numbers, Studia Sci. Math. Hungar. 18 (1983), 401–406.Google Scholar
[16] , On size Ramsey number of paths, trees and cycles I, J. Graph Theory 7 (1983), 115–130.Google Scholar
[17] , On size Ramsey number of paths, trees and cycles II, in Mathematics of Ramsey theory, Algorithms Combin., Vol. 5, 34–45, Springer, Berlin, 1990.
[18] , Achievement games and the probabilistic method, in Combinatorics, Paul Erdőos is Eighty, Vol. 1, 51–78, Bolyai Soc. Math. Stud., Jáanos Bolyai Math. Soc., Budapest, 1993.
[19] , and , Forcing large transitive subtournaments, to appear in J. Combin. Theory Ser. B.
[21] and , The early evolution of the H-free process, Invent. Math. 181 (2010), 291–336.Google Scholar
[22] and , Dynamic concentration of the trianglefree process, preprint.
[23] and , Graphs without large triangle free subgraphs, Discrete Math. 87 (1991), 119–131.Google Scholar
[24] , and , The Erdős–Hajnal conjecture for long holes and anti-holes, preprint.
[25] , and , The Erdőos–Hajnal conjecture for paths and antipaths, to appear in J. Combin. Theory Ser. B.
[26] , Expanding graphs and Ramsey numbers, available at Freie Universitäat, Berlin preprint server, ftp://ftp.math.fuberlin.de/pub/math/publ/pre/1996/pr-a-96-24.ps.
[27] and , Induced subgraphs of Ramsey graphs with many distinct degrees, J. Combin. Theory Ser. B 97 (2007), 612–619.Google Scholar
[28] , Ramsey numbers involving graphs with long suspended paths, J. London Math. Soc. 24 (1981), 405–413.Google Scholar
[29] , What can we hope to accomplish in generalized Ramsey theory?, Discrete Math. 67 (1987), 215–225.Google Scholar
[30] and , On the magnitude of generalized Ramsey numbers for graphs, in Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), 214–240, Colloq. Math. Soc. Janos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.
[31] and , Generalizations of a Ramsey-theoretic result of Chvátal, J. Graph Theory 7 (1983), 39–51.Google Scholar
[32] , , , and , Some complete bipartite graph-tree Ramsey numbers, Ann. Discrete Math. 41 (1989), 79–90.Google Scholar
[34] and , On the Ramsey multiplicity of graphs – problems and recent results, J. Graph Theory 4 (1980), 347–361.Google Scholar
[35] , Induced-universal graphs for graphs with bounded maximum degree, Graphs Combin. 25 (2009), 461–468.Google Scholar
[36] , , , , and , On-line Ramsey theory for bounded degree graphs, Electron. J. Combin. 18 (2011), P136.Google Scholar
[37] and , Graphs with linearly bounded Ramsey numbers, J. Combin. Theory Ser. B 57 (1993), 138–149.Google Scholar
[39] and , The Erdőos–Hajnal conjecture for bullfree graphs, J. Combin. Theory Ser. B 98 (2008), 1301–1310.Google Scholar
[40] and , Excluding paths and antipaths, to appear in Combinatorica.
[41] and , Erdőos on Graphs. His Legacy of Unsolved Problems, A K Peters, Ltd., Wellesley, MA, 1998.
[43] and , Generalized Ramsey theory for graphs. III. Small off-diagonal numbers, Pacific J. Math. 41 (1972), 335–345.Google Scholar
[44] , , and Jr, The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), 239–239.Google Scholar
[45] , A new upper bound for diagonal Ramsey numbers, Ann. of Math. 170 (2009), 941–941.Google Scholar
[46] , Hypergraph packing and sparse bipartite Ramsey numbers, Combin. Probab. Comput. 18 (2009), 913–913.Google Scholar
[48] , On the Ramsey multiplicity of complete graphs, Combinatorica 32 (2012), 171–171.Google Scholar
[50] , , and , Ramsey numbers of cubes versus cliques, to appear in Combinatorica.
[51] , , and , The Erdős-Gyárfás problem on generalized Ramsey numbers, Proc. Lond. Math. Soc. 110 (2015), 1–1.Google Scholar
[52] , , and , On the grid Ramsey problem and related questions, to appear in Int. Math. Res. Not.
[53] , , and , Ordered Ramsey numbers, submitted.
[54] , and , Ramsey numbers of sparse hyper-graphs, Random Structures Algorithms 35 (2009), 1–1.Google Scholar
[56] , and , An approximate version of Sidorenko's conjecture, Geom.Funct.Anal. 20 (2010), 1354–1354.Google Scholar
[57] , and , Large almost monochromatic subsets in hypergraphs, Israel J. Math. 181 (2011), 423–423.Google Scholar
[58] , and , On two problems in graph Ramsey theory, Combinatorial 32 (2012), 513–513.Google Scholar
[59] , and , Erdős-Hajnal-type theorems in hypergraphs, J. Combin. Theory Ser. B 102 (2012), 1142–1142.Google Scholar
[61] , and , Short proofs of some extremal results, Combin. Probab. Comput. 23 (2014), 8–8.Google Scholar
[62] , and , Short proofs of some extremal results II, in preparation.
[63] , and , Extremal results in sparse pseudo-random graphs, Adv. Math. 256 (2014), 206–206.Google Scholar
[64] and , Combinatorial theorems in sparse random sets, submitted.
[65] and , An upper bound for Folkman numbers, preprint.
[66] , , and , 3-uniform hy-pergraphs of bounded degree have linear Ramsey numbers, J. Combin. Theory Ser. B 98 (2008), 484–484.Google Scholar
[67] , , and , Embeddings and Ramsey numbers of sparse k-uniform hypergraphs, Combinatorica 28 (2009), 263–263.Google Scholar
[68] , D. Král', , , and , Monochromatic triangles in three-coloured graphs, J. Combin. Theory Ser. B 103 (2013), 489–489.Google Scholar
[69] , The size-Ramsey number of trees, Random Structures Algorithms 40 (2012), 49–49.Google Scholar
[70] , A generalization of Ramsey's theorem, in Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), 323–332, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.
[71] and , On generalized Ramsey numbers for 3-uniform hypergraphs, J. Graph Theory 76 (2014), 217–217.Google Scholar
[72] , and , On generalized Ramsey numbers of Erdős and Rogers, J. Combin. Theory Ser. B 109 (2014), 213–213.Google Scholar
[74] and , On Ks-free subgraphs in Ks+k-free graphs and vertex Folkman numbers, Combinatorica 31 (2011), 39–39.Google Scholar
[75] , , and , A fast approximation algorithm for computing the frequencies of subgraphs in a given graph, SIAM J. Comput. 24 (1995), 598–598.Google Scholar
[77] and , Edge-coloring cliques with many colors on subcliques, Combinatorica 20 (2000), 441–441.Google Scholar
[78] , Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–292.Google Scholar
[80] , On the number of complete subgraphs contained in certain graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962), 459–459.Google Scholar
[81] , Problems and results on finite and infinite graphs, in Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), 183–192, Academia, Prague, 1975.
[82] , Solved and unsolved problems in combinatorics and combinatorial number theory, in Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. I (Baton Rouge, La., 1981), Congr. Numer. 32 (1981), 49–49.Google Scholar
[83] , On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25–25.Google Scholar
[84] , On some problems in graph theory, combinatorial analysis and combinatorial number theory, in Graph theory and combinatorics (Cambridge, 1983), 1–17, Academic Press, London, 1984.
[85] , Problems and results on graphs and hypergraphs: similar-ities and differences, in Mathematics of Ramsey theory, 12-28, Algorithms Combin., 5, Springer, Berlin, 1990.
[86] , On some of my favourite problems in various branches of combinatorics, in Proceedings of the Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (Prachatice, 1990), 69–79, Ann. Discrete Math., 51, North-Holland, Amsterdam, 1992.
[87] , Problems and results in discrete mathematics, Discrete Math. 136 (1994), 53–53.Google Scholar
[89] and , On partition theorems for finite graphs, in Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), 515–527, Colloq. Math.Soc.Jaanos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.
[90] and , A variant of the classical Ramsey problem, Combinatorica 17 (1997), 459–459.Google Scholar
[92] and , On Ramsey like theorems, problems and results, in Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), 123–140, Inst. Math. Appl., Southend-on-Sea, 1972.
[93] and , On spanned subgraphs of graphs, in Contributions to graph theory and its applications (Internat. Colloq., Oberhof, 1977), 80–96, Tech. Hochschule Ilmenau, Ilmenau, 1977.Google Scholar
[95] , and , On a metric generalization of Ram-sey's theorem, Israel J. Math. 102 (1997), 283–283.Google Scholar
[96] , and , A Ramsey-type theorem for bipartite graphs, Geombinatorics 10 (2000), 64–64.Google Scholar
[97] , and , Strong embeddings of graphs into colored graphs, in Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), 585–595, Colloq. Math. Soc. Jaanos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.
[98] , and , Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–93.Google Scholar
[99] and , Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. 3 (1952), 417–417.Google Scholar
[101] and , Cube-supersaturated graphs and related problems, in Progress in graph theory (Waterloo, Ont., 1982), 203–218, Academic Press, Toronto, ON, 1984.
[104] , and , The triangle-free process and R(3,k), preprint.
[105] , , , and , On the Ramsey number of the triangle and the cube, to appear in Combinatorica.
[106] , , , and , The Ramsey number of the clique and the hypercube, J. Lond. Math. Soc. 89 (2014), 680–680.Google Scholar
[107] , Graphs with monochromatic complete subgraphs in every edge coloring, SIAM J. Appl. Math. 18 (1970), 19–19.Google Scholar
[108] , There exist graphs with super-exponential Ramsey multiplicity constant, J. Graph Theory 57 (2008), 89–89.Google Scholar
[109] , and , The Erdős-Hajnal conjecture for rainbow triangles, J. Combin. Theory Ser. B 111 (2015), 75–75.Google Scholar
[110] and , Erdős–Hajnal-type results on intersection patterns of geometric objects, in Horizons of combinatorics, 79–103, Bolyai Soc. Math. Stud., 17, Springer, Berlin, 2008.
[111] and , Applications of a new separator theorem for string graphs, Combin. Probab. Comput. 23 (2014), 66–66.Google Scholar
[112] , , and , Erdős-Szekeres-type theorems for monotone paths and convex bodies, Proc. Lond. Math. Soc. 105 (2012), 953–953.Google Scholar
[113] , and , Intersection patterns of curves, J. Lond. Math. Soc. 83 (2011), 389–389.Google Scholar
[115] and , Ramsey-type problem for an almost monochromatic K4, SIAM J. Discrete Math. 23 (2008/09), 155-162.
[116] and , Density theorems for bipartite graphs and related Ramsey-type results, Combinatorica 29 (2009), 153–153.Google Scholar
[117] and , Two remarks on the Burr-Erdős conjecture, European J. Combin. 30 (2009), 1630–1630.Google Scholar
[119] and , Large triangle-free subgraphs in graphs with-out K4, Graphs Combin. 2 (1986), 135–135.Google Scholar
[120] and , Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357–357.Google Scholar
[121] , and , Ramsey properties of discrete random structures, Random Structures Algorithms 37 (2010), 407–407.Google Scholar
[123] and , Algebraic Graph Theory, Springer, 2001.
[124] , On sets of acquaintances and strangers at any party, Amer. Math. Monthly 66 (1959), 778–778.Google Scholar
[125] , A new proof of Szemeraédi's theorem for arithmetic progressions of length four, Geom.Funct.Anal. 8 (1998), 529–529.Google Scholar
[126] , Hypergraph regularity and the multidimensional Szemeraedi theorem, Ann. of Math. 166 (2007), 897–897.Google Scholar
[127] and , Numbers in Ramsey theory, in Surveys in Combinatorics 1987, 111–153, London Math. Soc. Lecture Note Ser., Vol. 123, Cambridge University Press, Cambridge, 1987.
[128] , and , On graphs with linear Ramsey numbers, J. Graph Theory 35 (2000), 176–176.Google Scholar
[129] , and , On bipartite graphs with linear Ramsey numbers, Combinatorica 21 (2001), 199–199.Google Scholar
[130] , and , Ramsey theory, 2nd edition, Wiley, 1990.
[131] and , The primes contain arbitrarily long arithmetic progressions, Ann. of Math. 167 (2008), 481–481.Google Scholar
[132] , and , On-line Ramsey theory, Electron. J. Combin. 11 (2004), Research Paper 60, 10pp.Google Scholar
[133] , On a Ramsey type problem of Shelah, in Extremal prob-lems for finite sets (Visegrád, 1991), 283–287, Bolyai Soc. Math. Stud., 3, Jaanos Bolyai Math. Soc., Budapest, 1994.
[134] , Rainbow Ramsey theorems for colorings establishing negative partition relations, Fund. Math. 198 (2008), 255–255.Google Scholar
[135] and , A mathematical incompleteness in Peano arithmetic, in Handbook of Mathematical Logic, 1133–1142, North-Holland, Amsterdam, 1977.
[136] , , D. Král', and , Non-three-colorable common graphs exist, Combin. Probab. Comput. 21 (2012), 734–734.Google Scholar
[137] , and , The induced size-Ramsey number of cycles, Combin. Probab. Comput. 4 (1995), 217–217.Google Scholar
[138] , and , Degree Ramsey numbers of closed blowups of trees, Electron. J. Combin. 21 (2014), Paper 2.5, 6pp.Google Scholar
[140] and , Coloring number and on-line Ramsey theory for graphs and hypergraphs, Combinatorica, 29 (2009), 49–49.Google Scholar
[141] , The Ramsey number R(3, t) has order of magnitude t2/ log t, Random Structures Algorithms 7 (1995), 173–173.Google Scholar
[142] , and , Two approaches to Sidorenko's conjecture, to appear in Trans. Amer. Math. Soc.
[143] , and , Degree Ramsey numbers of graphs, Combin. Probab. Comput. 21 (2012), 229–229.Google Scholar
[145] , , and , Sparse partition universal graphs for graphs of bounded degree, Adv. Math. 226 (2011), 5041–5041.Google Scholar
[146] and , When is an almost monochromatic K4 guaranteed?Combin. Probab. Comput. 17 (2008), 823–823.Google Scholar
[148] and , On Ramsey numbers of uniform hy-pergraphs with given maximum degree, J. Combin. Theory Ser. A 113 (2006), 1555–1555.Google Scholar
[149] and , On Ramsey numbers of sparse graphs, Combin. Probab. Comput. 12 (2003), 627–627.Google Scholar
[150] , Bounding Ramsey numbers through large deviation inequalities, Random Structures Algorithms 7 (1995), 145–145.Google Scholar
[151] and , Pseudo-random graphs, in More sets, graphs and numbers, 199–262, Bolyai Soc. Math. Stud., 15, Springer, Berlin, 2006.
[152] and , Two variants of the size Ramsey number, Discuss. Math. Graph Theory 25 (2005), 141–141.Google Scholar
[153] , and , Use of MAX-CUT for Ramsey arrowing of triangles, J. Combin. Math. Combin. Comput. 88 (2014), 61–61.Google Scholar
[154] , , and , A Ramsey-type result for convex sets, Bull. London Math. Soc. 26 (1994), 132–132.Google Scholar
[155] and , On the logarithmic calculus and Sidorenko's conjecture, to appear in Combinatorica.
[156] , Explicit construction of small Folkman graphs, SIAM J. Discrete Math. 21 (2007), 1053–1053.Google Scholar
[157] and , On induced Ramsey numbers for graphs with bounded maximum degree, J. Combin. Theory Ser. B 66 (1996), 324–324.Google Scholar
[159] , and , Ordered Ramsey theory and track representations of graphs, to appear in J. Combin.
[160] , Ramsey–Paris–Harrington numbers for graphs, J. Combin. Theory Ser. A 38 (1985), 30–30.Google Scholar
[161] and , Ramsey theory, integer partitions and a new proof of the Erdős-Szekeres theorem, Adv. Math. 262 (2014), 1107–1107.Google Scholar
[162] , Edge-coloring cliques with three colors on all 4-cliques, Combinatorica 18 (1998), 293–293.Google Scholar
[163] , , and , On the Ramsey number of sparse 3-graphs, Graphs Combin. 27 (2008), 205–205.Google Scholar
[164] , and , The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms 28 (2006), 113–179.Google Scholar
[165] and , A short proof of the random Ramsey theorem, to appear in Combin. Probab. Comput.
[166] and , The Ramsey property for graphs with forbidden complete subgraphs, J. Combin. Theory Ser. B 20 (1976), 243–243.Google Scholar
[167] and , Simple proof of the existence of restricted Ramsey graphs by means of a partite construction, Combinatorica 1 (1981), 199–199.Google Scholar
[168] and , Combinatorics and quantifiers, Comment. Math. Univ. Carolin. 37 (1996), 433–433.Google Scholar
[169] , Counting monochromatic copies of K4: a new lower bound for the Ramsey multiplicity problem, preprint.
[171] and , The chromatic Ramsey number of odd wheels, J. Graph Theory 69 (2012), 198–198.Google Scholar
[172] and , Non-Ramsey graphs are c log n-universal, J. Combin. Theory Ser. A 88 (1999), 379–379.Google Scholar
[173] , Small Ramsey numbers, Electron. J. Combin. (2014), DS1.
[176] , The dimension of a graph and generalized Ramsey theorems, Master's thesis, Charles University, 1973.
[177] , On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), 125–125.Google Scholar
[178] , On homogeneous sets of positive integers, J. Combin. Theory Ser. A 102 (2003), 229–229.Google Scholar
[179] and , Lower bounds on probability thresholds for Ramsey properties, in Combinatorics, Paul Erdős is eighty, Vol. 1, 317–346, Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest, 1993.
[180] and , Threshold functions for Ramsey properties, J. Amer. Math. Soc. 8 (1995), 917–917.Google Scholar
[181] , and , An exponential-type upper bound for Folkman numbers, preprint.
[182] and , Complete partite subgraphs in dense hypergraphs, Random Structures Algorithms 41 (2012), 557–557.Google Scholar
[183] and , Regularity lemma for uniform hypergraphs, Random Structures Algorithms 25 (2004), 1–1.Google Scholar
[184] and , On size Ramsey numbers of graphs with bounded maximum degree, Combinatorica 20 (2000), 257–257.Google Scholar
[185] and , Arrangeability and clique subdivisions, in The mathematics of Paul Erdős, II, 236–239, Algorithms Combin., 14, Springer, Berlin, 1997.Google Scholar
[186] and , On edge colorings with at least q colors in every subset of p vertices, Electron.J.Combin. 8 (2001), Research Paper 9, 6pp.Google Scholar
[187] and , An application of the regularity lemma in generalized Ramsey theory, J. Graph Theory 44 (2003), 39–39.Google Scholar
[188] and , Hypergraph containers, to appear in Invent. Math.
[189] , Extremal results for discrete random structures, preprint.
[190] , A note on the independence number of triangle-free graphs, Discrete Math. 46 (1983), 83–83.CrossRefGoogle Scholar
[191] , On the independence number of sparse graphs, Random Structures Algorithms 7 (1995), 269–269.Google Scholar
[192] , Primitive recursive bounds for van der Waerden numbers, J. Amer. Math. Soc. 1 (1989), 683–683.CrossRefGoogle Scholar
[193] , A finite partition theorem with double exponential bound, in The mathematics of Paul Erdős, II, 240–246, Algorithms Combin., 14, Springer, Berlin, 1997.
[194] , Cube Ramsey numbers are polynomial, Random Structures Algorithms 19 (2001), 99–99.Google Scholar
[195] , The tail is cut for Ramsey numbers of cubes, Discrete Math. 307 (2007), 290–290.CrossRefGoogle Scholar
[196] , Cycles in graphs and functional inequalities, Math. Notes 46 (1989), 877–877.CrossRefGoogle Scholar
[197] , A correlation inequality for bipartite graphs, Graphs Combin. 9 (1993), 201–201.CrossRefGoogle Scholar
[198] , Ramsey's theorem – a new lower bound, J. Combin. Theory Ser. A 18 (1975), 108–108.Google Scholar
[199] , Asymptotic lower bounds for Ramsey functions, Discrete Math. 20 (1977/78), 69–76.Google Scholar
[200] , Three hundred million points suffice, J. Combin. The-ory Ser. A 49 (1988), 210–210. See also the erratum by M. Hovey in J. Combin. Theory Ser. A 50 (989), 323.Google Scholar
[201] , On the minimal monochromatic K4-density, preprint.
[202] , A few remarks on the Ramsey-Turán-type problems, J. Combin. Theory Ser. B 88 (2003), 99–99.Google Scholar
[203] , A new lower bound for a Ramsey-type problem, Combinatorica 25 (2005), 487–487.Google Scholar
[204] , Large Kr-free subgraphs in Ks-free graphs and some other Ramsey-type problems, Random Structures Algorithms 26 (2005), 253–253.Google Scholar
[205] , A conjecture of Erdős on graph Ramsey numbers, Adv. Math. 227 (2011), 601–601.Google Scholar
[206] , On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–199.Google Scholar
[207] , Regular partitions of graphs, in Problémes Combinatoires et Théorie des Graphes (Orsay 1976), 399–401, Colloq. Internat. CNRS, 260, CNRS, Paris, 1978.
[208] , Pseudorandom graphs, in Random graphs '85 (Poznań, 1985), 307–331, North-Holland Math. Stud., Vol. 144, NorthHolland, Amsterdam, 1987.
[209] , Random graphs, strongly regular graphs and pseudorandom graphs, in Surveys in Combinatorics 1987, 173–195, London Math. Soc. Lecture Note Ser., Vol. 123, Cambridge University Press, Cambridge, 1987.
[211] , A disproof of a conjecture of Erdos in Ramsey theory, J. London Math. Soc. 39 (1989), 246–246.Google Scholar
[213] , K4-free graphs without large induced triangle-free subgraphs, Combinatorica 33 (2013), 623–623.Google Scholar
[214] , The fractional version of Hedetniemi's conjecture is true, European J. Combin. 32 (2011), 1168–1168.Google Scholar
- 42
- Cited by