Essays in extremal combinatorics

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.

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

[2]  E. Szemeri~di,et al.  On Sets of Integers Containing No Four Elements in Arithmetic Progression , .

[3]  Robin Thomas,et al.  Large induced forests in sparse graphs , 2001, J. Graph Theory.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[19]  Terence Tao,et al.  A sharp inverse Littlewood‐Offord theorem , 2009, Random Struct. Algorithms.

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

[21]  Benny Sudakov,et al.  Large Kr‐free subgraphs in Ks‐free graphs and some other Ramsey‐type problems , 2005, Random Struct. Algorithms.

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

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

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

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

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

[27]  J. Orlin.  Contentment in graph theory: Covering graphs with cliques , 1977 .

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

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

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

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

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

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

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

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

[36]  Van H. Vu.  Inverse Littlewood-Offord theorems and the condition number of random discrete matrices , 2009 .

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

[38]  Yoshiharu Kohayakawa,et al.  Ramsey Games Against a One-Armed Bandit , 2003, Comb. Probab. Comput..

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

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

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

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

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

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

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

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