The New Combinatorica

Combinatorica: A System for Exploring Combinatorics and Graph Theory in Mathematica

Combinatorica is an extension to the computer algebra system Mathematica [9] that provides over 450 functions for discrete mathematics. It is distributed as a standard package with every copy of Mathematica. Combinatorica facilitates the counting, enumeration, visualization, and manipulation of permutations, combinations, integer and set partitions, Young tableaux, partially ordered sets, trees...

A Pushing-Pulling Method: New Proofs of Intersection Theorems

A new approach to the Ramsey-type games and the Gowers dichotomy in F-spaces

We give a new approach to the Ramsey-type results of Gowers on block bases in Banach spaces and apply our results to prove the Gowers dichotomy in F-spaces.

New bounds in Balog-Szemerédi-Gowers theorem

We prove, in particular, that every finite subset A of an abelian group with the additive energy κ|A| contains a set A′ such that |A′| ≫ κ|A| and |A′ −A′| ≪ κ−4|A′|.

A new lower bound for snake-in-the-box codes

In this paper we give a new lower bound on the length of Snake-in-the-Box Codes, i.e., induced cycles in the d-dimensional cube. The bound is a linear function of the number of vertices of the cube and improves the bound c · 2/d, where c is a constant, proved by Danzer and Klee. AMS subject classification 1980: 05 C 35, 94 B 25