Short proofs of some extremal results II

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

[1]  Choongbum Lee Ramsey numbers of degenerate graphs , 2015, 1505.04773.

[2]  Yoshiharu Kohayakawa,et al.  Induced Ramsey Numbers , 1998, Comb..

[3]  Benny Sudakov,et al.  A New Lower Bound For A Ramsey-Type Problem , 2005, Comb..

[4]  A. Goodman On Sets of Acquaintances and Strangers at any Party , 1959 .

[5]  Jacob Fox,et al.  Complete Minors and Independence Number , 2010, SIAM J. Discret. Math..

[6]  Jane Butterfield,et al.  On-line Ramsey Theory for Bounded Degree Graphs , 2011, Electron. J. Comb..

[7]  Benny Sudakov,et al.  Two remarks on the Burr-Erdos conjecture , 2009, Eur. J. Comb..

[8]  Benny Sudakov,et al.  Induced Ramsey-type theorems , 2007, Electron. Notes Discret. Math..

[9]  Hal A. Kierstead,et al.  Coloring number and on-line Ramsey theory for graphs and hypergraphs , 2009, Comb..

[10]  David Conlon,et al.  Hedgehogs are not colour blind , 2015 .

[11]  Béla Bollobás,et al.  Random Graphs , 1985 .

[12]  T. Tao,et al.  Inverse Littlewood-Offord theorems and the condition number of random discrete matrices , 2005, math/0511215.

[13]  Lutz Warnke,et al.  On the Method of Typical Bounded Differences , 2012, Combinatorics, Probability and Computing.

[14]  Ingo Schiermeyer,et al.  The Ramsey number r(C7, C7, C7) , 2003, Discuss. Math. Graph Theory.

[15]  Vojtech Rödl,et al.  The Ramsey number of a graph with bounded maximum degree , 1983, J. Comb. Theory, Ser. B.

[16]  David Conlon,et al.  The Ramsey number of dense graphs , 2009, 0907.2657.

[17]  Svante Janson,et al.  The Chromatic Number , 2011 .

[18]  Andrzej Dudek,et al.  On Ks-free subgraphs in Ks+k-free graphs and vertex Folkman numbers , 2011, Comb..

[19]  Béla Bollobás,et al.  Large joints in graphs , 2010, Eur. J. Comb..

[20]  R. Salem,et al.  On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[21]  N. Alon Problems and results in Extremal Combinatorics – III , 2016 .

[22]  Richard H. Schelp,et al.  Ramsey unsaturated and saturated graphs , 2006 .

[23]  O. Ore Theory of Graphs , 1962 .

[24]  Benny Sudakov,et al.  Dependent random choice , 2009, Random Struct. Algorithms.

[25]  Imre Z. Ruzsa,et al.  An Infinite Sidon Sequence , 1998 .

[26]  Angelika Steger,et al.  Online Ramsey Games in Random Graphs , 2009, Comb. Probab. Comput..

[27]  Paul Erdös,et al.  On the Difference between Consecutive Ramsey Numbers , 1989 .

[28]  Yury Person,et al.  On the multicolor Ramsey number of a graph with m edges , 2016, Discret. Math..

[29]  Joel H. Spencer,et al.  Turán's theorem for k-graphs , 1972, Discret. Math..

[30]  Fred B. Schneider,et al.  A Theory of Graphs , 1993 .

[31]  N. Hegyvári On the Dimension of the Hilbert Cubes , 1999 .

[32]  David Conlon,et al.  On-line Ramsey Numbers , 2009, SIAM J. Discret. Math..

[33]  D. Conlon Combinatorial theorems relative to a random set , 2014, 1404.3324.

[34]  Dhruv Mubayi,et al.  Books Versus triangles , 2012, J. Graph Theory.

[35]  R. A. Sáenz,et al.  Problems , 2017 .

[36]  S. Burr ON THE MAGNITUDE OF GENERALIZED RAMSEY NUMBERS FOR GRAPHS , 1973 .

[37]  Paul Erdös,et al.  Quantitative Forms of a Theorem of Hilbert , 1985, J. Comb. Theory, Ser. A.

[38]  A. Kostochka The minimum Hadwiger number for graphs with a given mean degree of vertices , 1982 .

[39]  J. Kahn,et al.  On the number of copies of one hypergraph in another , 1998 .

[40]  Jeong Han Kim,et al.  The Ramsey Number R(3, t) Has Order of Magnitude t2/log t , 1995, Random Struct. Algorithms.

[41]  K. F. Roth On Certain Sets of Integers , 1953 .

[42]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[43]  Béla Bollobás,et al.  Books in graphs , 2005, Eur. J. Comb..

[44]  F. Behrend On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1946, Proceedings of the National Academy of Sciences of the United States of America.

[45]  Andrzej Dudek,et al.  On Generalized Ramsey Numbers for 3‐Uniform Hypergraphs , 2013, J. Graph Theory.

[46]  W. Mader Homomorphieeigenschaften und mittlere Kantendichte von Graphen , 1967 .

[47]  Van Vu,et al.  Optimal Inverse Littlewood-Offord theorems , 2010, 1004.3967.

[48]  Noga Alon,et al.  Problems and results in extremal combinatorics--I , 2003, Discret. Math..

[49]  D. Hilbert,et al.  Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten , 1933 .

[50]  J. Sheehan,et al.  On the number of complete subgraphs contained in certain graphs , 1981, J. Comb. Theory, Ser. B.

[51]  Mohammad R. Salavatipour Large Induced Forests in Triangle-Free Planar Graphs , 2006, Graphs Comb..

[52]  David Conlon,et al.  Recent developments in graph Ramsey theory , 2015, Surveys in Combinatorics.

[53]  Vojtech Rödl,et al.  Extremal Problems for Sets Forming Boolean Algebras and Complete Partite Hypergraphs , 1999, J. Comb. Theory, Ser. A.

[54]  I. Shkredov On a Generalization of Szemerédi's Theorem , 2005, math/0503639.

[55]  A. Thomason An extremal function for contractions of graphs , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[56]  Vojtech Rödl,et al.  On graphs with linear Ramsey numbers , 2000, J. Graph Theory.

[57]  Zoltán Füredi Maximal Independent Subsets in Steiner Systems and in Planar Sets , 1991, SIAM J. Discret. Math..

[58]  Angelika Steger,et al.  Upper Bounds for Online Ramsey Games in Random Graphs , 2009, Comb. Probab. Comput..

[59]  David Conlon,et al.  On two problems in graph Ramsey theory , 2012, Comb..

[60]  Paul Erdös,et al.  The Construction of Certain Graphs , 1966, Canadian Journal of Mathematics.

[61]  Paul Erdös,et al.  Ramsey-type theorems , 1989, Discret. Appl. Math..

[62]  Riste Skrekovski,et al.  An improved bound on the largest induced forests for triangle-free planar graphs , 2010, Discret. Math. Theor. Comput. Sci..

[63]  Lingsheng Shi The tail is cut for Ramsey numbers of cubes , 2007, Discret. Math..

[64]  Jaroslav Nesetril,et al.  Sparsity - Graphs, Structures, and Algorithms , 2012, Algorithms and combinatorics.

[65]  Jaroslav Nesetril,et al.  Rainbow Ramsey Theory , 2005 .

[66]  M. A. Perles ON THE NUMBER OF SUBGRAPHS OF PRESCRIBED TYPE OF GRAPHS WITH A GIVEN NUMBER OF EDGES* , 2007 .

[67]  On representation problems in the additive number theory , 1996 .

[68]  Benny Sudakov,et al.  Density theorems for bipartite graphs and related Ramsey-type results , 2007, Comb..

[69]  D. Conlon A new upper bound for diagonal Ramsey numbers , 2006, math/0607788.

[70]  A. Thomason A Disproof of a Conjecture of Erdős in Ramsey Theory , 1989 .

[71]  Jin Akiyama,et al.  Maximum induced forests of planar graphs , 1987, Graphs Comb..

[72]  J. Solymosi Note on a Generalization of Roth’s Theorem , 2003 .

[73]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[74]  E. Szemerédi On sets of integers containing no four elements in arithmetic progression , 1969 .

[75]  Benny Sudakov Large K r -free subgraphs in K s -free graphs and some other Ramsey-type problems , 2005 .

[76]  Noga Alon,et al.  The concentration of the chromatic number of random graphs , 1997, Comb..

[77]  Richard Montgomery,et al.  Logarithmically small minors and topological minors , 2013, J. Lond. Math. Soc..

[78]  Andras Hajnal,et al.  On the structure of set-mappings , 1958 .

[79]  Michael Krivelevich,et al.  Bounding Ramsey Numbers through Large Deviation Inequalities , 1995, Random Struct. Algorithms.

[80]  P. Erdös On extremal problems of graphs and generalized graphs , 1964 .

[81]  Alexandr V. Kostochka,et al.  Lower bound of the hadwiger number of graphs by their average degree , 1984, Comb..

[82]  Andrew Thomason,et al.  The Extremal Function for Complete Minors , 2001, J. Comb. Theory B.

[83]  Béla Bollobás,et al.  Joints in graphs , 2008, Discret. Math..

[84]  D. Conlon,et al.  Hypergraph Ramsey numbers , 2008, 0808.3760.

[85]  P. Duchet,et al.  On Hadwiger''s number and stability number , 1982 .

[86]  János Komlós,et al.  A Note on Ramsey Numbers , 1980, J. Comb. Theory, Ser. A.

[87]  Hal A. Kierstead,et al.  On-line Ramsey Theory , 2004, Electron. J. Comb..

[88]  Benny Sudakov,et al.  Small complete minors above the extremal edge density , 2015, Comb..

[89]  Maya Jakobine Stein,et al.  Cycles Are Strongly Ramsey-Unsaturated , 2012, Combinatorics, Probability and Computing.

[90]  P. Erdös,et al.  Combinatorial Theorems on Classifications of Subsets of a Given Set , 1952 .

[91]  Andrzej Rucinski,et al.  Two variants of the size Ramsey number , 2005, Discuss. Math. Graph Theory.

[92]  Michael Krivelevich,et al.  Ks-Free Graphs Without Large Kr-Free Subgraphs , 1994, Combinatorics, Probability and Computing.

[93]  József Solymosi,et al.  Monochromatic Equilateral Right Triangles on the Integer Grid , 2006 .

[94]  Ken-ichi Kawarabayashi,et al.  Improvements of the theorem of Duchet and Meyniel on Hadwiger's conjecture , 2005, J. Comb. Theory, Ser. B.

[95]  Shi Lingsheng Cube Ramsey numbers are polynomial , 2001, Random Struct. Algorithms.

[96]  Richard M. Wilson,et al.  Constructions and Uses of Pairwise Balanced Designs , 1975 .

[97]  David Conlon,et al.  Short Proofs of Some Extremal Results , 2014, Comb. Probab. Comput..

[98]  Jacob Fox There exist graphs with super-exponential Ramsey multiplicity constant , 2008 .

[99]  Yoshiharu Kohayakawa,et al.  The Induced Size-Ramsey Number of Cycles , 1995, Combinatorics, Probability and Computing.

[100]  Thomas F. Bloom,et al.  A quantitative improvement for Roth's theorem on arithmetic progressions , 2014, J. Lond. Math. Soc..

[101]  Yanxun Chang,et al.  A bound for Wilson's theorem (III) , 1996 .

[102]  Vojtech Rödl,et al.  On Bipartite Graphs with Linear Ramsey Numbers , 2001, Comb..

[103]  J. Rassias Solution of a problem of Ulam , 1989 .

[104]  Terence Tao,et al.  A sharp inverse Littlewood-Offord theorem , 2010 .

[105]  Benny Sudakov,et al.  Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz , 2013, Comb..

[106]  Paul Erdös,et al.  On a theorem of Rademacher-Turán , 1962 .

[107]  Noga Alon,et al.  Constructive Bounds for a Ramsey-Type Problem , 1997, Graphs Comb..

[108]  David Conlon On the Ramsey multiplicity of complete graphs , 2012, Comb..

[109]  Noga Alon,et al.  Large induced forests in sparse graphs , 2001 .

[110]  P. Erdos,et al.  On a Ramsey type theorem , 1972 .

[111]  Andrzej Dudek,et al.  On generalized Ramsey numbers of Erdős and Rogers , 2014, J. Comb. Theory, Ser. B.

[112]  Jacques Verstraëte,et al.  Clique partitions of complements of forests and bounded degree graphs , 2008, Discret. Math..

[113]  Jan Hladký,et al.  Turánnical hypergraphs , 2010, Random Struct. Algorithms.

[114]  David A. Gregory,et al.  Clique partitions of the cocktail party graph , 1986, Discret. Math..

[115]  P. Erdös On an extremal problem in graph theory , 1970 .

[116]  Jacob Fox,et al.  A new proof of the graph removal lemma , 2010, ArXiv.

[117]  Michael Krivelevich,et al.  Large Subgraphs without Short Cycles , 2014, SIAM J. Discret. Math..

[118]  Nancy Eaton Ramsey numbers for sparse graphs , 1998, Discret. Math..

[119]  Noga Alon,et al.  Sub-Ramsey numbers for arithmetic progressions , 1989, Graphs Comb..

[120]  Noga Alon,et al.  Extremal Problems Concerning Transformations of the Set of Edges of the Complete Graph , 1986, Eur. J. Comb..

[121]  Noga Alon,et al.  Coloring Graphs with Sparse Neighborhoods , 1999, J. Comb. Theory B.

[122]  Yair Caro Extremal problems concerning transformations of the edges of the complete hypergraphs , 1987, J. Graph Theory.

[123]  Zoltán Füredi,et al.  A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity , 2015, J. Comb. Theory, Ser. B.

[124]  Satish Rao,et al.  Shallow excluded minors and improved graph decompositions , 1994, SODA '94.

[125]  Zoltán Füredi,et al.  Connected matchings and Hadwiger's conjecture , 2005, Combinatorics, Probability and Computing.

[126]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[127]  E. Szemerédi Regular Partitions of Graphs , 1975 .

[128]  David R. Wood,et al.  Small minors in dense graphs , 2010, Eur. J. Comb..

[129]  T. Sanders On Roth's theorem on progressions , 2010, 1011.0104.

[130]  Pál Erdös On the number of complete subgraphs and circuits contained in graphs , 1969 .

[131]  Benny Sudakov A conjecture of Erdős on graph Ramsey numbers , 2011 .

[132]  H. Davenport Multiplicative Number Theory , 1967 .

[133]  James B. Shearer,et al.  On the Independence Number of Sparse Graphs , 1995, Random Struct. Algorithms.

[134]  David Conlon Hypergraph Packing and Sparse Bipartite Ramsey Numbers , 2009, Comb. Probab. Comput..

[135]  Noga Alon,et al.  Problems and results in extremal combinatorics - II , 2008, Discret. Math..