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.

In this work we study numbers and polynomials generated by two type of composition of generating functions and get their explicit formulae. Furthermore we state an improvementof the composita formulae's given in [6] and [3], using the new composita formula's we construct a variety of combinatorics identities. This study go alone to dene new family of generalized Bernoulli polynomials which incl...

Generalizing the concept of zero{error capacity beyond its traditional links to any sort of information transmission we give an asymptotic solution to several hard problems in extremal set theory within a uniied, formally information{theoretic framework. The results include the solution of far{reaching generalizations of R enyi's problem on qualitatively independent partitions.

Higher incidence matrices have proved an important tool both in design theory and extremal set theory. In the present paper some tight bounds on the rank over finite fields of some inclusion matrices are derived. In particular, a short proof of Wilson's mod p rank formula is given. A problem of Graham, Li, and Li concerning bases for so-called null t-designs is solved as well.

A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain F1 ⊂ F2. Erdős extended this theorem to determine the largest family without a k-chain F1 ⊂ F2 ⊂ . . . ⊂ Fk. Erdős and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the c...

Theorems on extremal set systems with intersections of restricted cardinality are probably among the most well-known and powerful theorems in extremal combinatorics. In this paper, we discuss the so-call ”weak” version of some theorems of this type, when the restricted intersection property is weakened by the possible existence of some (maybe many) intersections having ”bad” sizes. In particula...

These notes arose from my Cambridge Part III course on Additive Combinatorics, given in Lent Term 2009. The aim was to understand the simplest proof of the Bourgain-Glibichuk-Konyagin bounds for exponential sums over subgroups. As a byproduct one obtains a clean proof of the Bourgain-Katz-Tao theorem on the sumproduct phenomenon in Fp. The arguments are essentially extracted from Bourgain’s pap...

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains Ω(n) points of P and no circle contains Ω(n) points of P . We rely on the bipartite and partial variant of the Elekes-Sharir framework that was presented by Sharir, Sheffer, and Solymosi in [2...

Der Einsatz von Audience Response Systemen ermöglicht Hochschullehrenden die Interaktion mit Studierenden in Lehrveranstaltungen mit vielen Teilnehmenden. Seit einem Jahr steht allen Dozierenden und Studierenden der Goethe-Universität das softwarebasierte Audience Response System eduVote zur Verfügung. Im Rahmen einer empirischen Studie wurden erste Erfahrungen der Lehrenden mit dem System unte...