Set-polynomials and polynomial extension of the Hales-Jewett theorem.
Sets of points determining only acute angles and some related colouring problems.
Set-systems with restricted multiple intersections.
Small forbidden configurations. II.
Small forbidden configurations. III.
Some combinatorics involving ξ-large sets
We prove a version of the Ramsey theorem for partitions of (increasing) n-tuples. We derive this result from a version of König's infinity lemma for ξ-large trees. Here ξ < ε₀ and the notion of largeness is in the sense of Hardy hierarchy.
Some Geometric Applications of Dilworth's Theorem.
Some new exact van der Waerden numbers.
Some new Ramsey colorings.
Some Ramsey type theorems for normed and quasinormed spaces
We prove that every bounded, uniformly separated sequence in a normed space contains a “uniformly independent” subsequence (see definition); the constants involved do not depend on the sequence or the space. The finite version of this result is true for all quasinormed spaces. We give a counterexample to the infinite version in for each 0 < p < 1. Some consequences for nonstandard topological vector spaces are derived.
Some results in additive number theory I: The critical pair theory
Some results on the problem of Erdös and Hajnal.
Stability analysis for -wise intersecting families.
Standard monomials for q-uniform families and a conjecture of Babai and Frankl
Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family We study certain inclusion matrices attached to F(k,q) over the field . We show that if l≤q−1 and 2l≤n then This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.
Strings with maximally many distinct subsequences and substrings.
Strongly cancellative and recovering sets on lattices.
Subgraph densities in hypergraphs
Let r ≥ 2 be an integer. A real number α ∈ [0,1) is a jump for r if for any ε > 0 and any integer m ≥ r, any r-uniform graph with n > n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c = c(α) > 0 does not depend on ε and m. A result of Erdös, Stone and Simonovits implies that every α ∈ [0,1) is a jump for r = 2. Erdös asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence...
Symmetric sum-free partitions and lower bounds for Schur numbers.
Symmetrized and continuous generalization of transversals
The theorem of Edmonds and Fulkerson states that the partial transversals of a finite family of sets form a matroid. The aim of this paper is to present a symmetrized and continuous generalization of this theorem.
The 11-element case of Frankl's conjecture.