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On families of weakly dependent random variables

Tomasz Łuczak — 2011

Banach Center Publications

Let $ₙ^{(k)}$ be a family of random independent k-element subsets of [n] = 1,2,...,n and let $(ₙ^{(k)}, ℓ) = ₙ^{(k)} (ℓ)$ denote a family of ℓ-element subsets of [n] such that the event that S belongs to $ₙ^{(k)} (ℓ)$ depends only on the edges of $ₙ^{(k)}$ contained in S. Then, the edges of $ₙ^{(k)} (ℓ)$ are ’weakly dependent’, say, the events that two given subsets S and T are in $ₙ^{(k)} (ℓ)$ are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We also list...

On the maximum automorphism group of self-complementary graphs

Tomasz Łuczak — 2000

Mathematica Slovaca

On random subsets of projective spaces

Wojciech Kordecki; Tomasz Łuczak — 1991

Colloquium Mathematicae

Small bases for finite groups

Tomasz Łuczak; Tomasz Schoen — 1998

Colloquium Mathematicae

On strongly sum-free subsets of abelian groups

Tomasz Łuczak; Tomasz Schoen — 1996

Colloquium Mathematicae

In his book on unsolved problems in number theory [1] R. K. Guy asks whether for every natural l there exists $n_{0} = n_{0} (l)$ with the following property: for every $n \geq n_{0}$ and any n elements $a_{1}, . . ., a_{n}$ of a group such that the product of any two of them is different from the unit element of the group, there exist l of the $a_{i}$ such that $a_{i_{j}} a_{i_{k}} \neq a_{m}$ for $1 \leq j < k \leq l$ and $1 \leq m \leq n$ . In this note we answer this question in the affirmative in the first non-trivial case when l=3 and the group is abelian, proving the following result.

On the maximal density of sum-free sets

Tomasz Łuczak; Tomasz Schoen — 2000

Acta Arithmetica

On a problem of Konyagin

Tomasz Łuczak; Tomasz Schoen — 2008

Acta Arithmetica

Arithmetic progressions of length three in subsets of a random set

Yoshiharu Kohayakawa; Tomasz Łuczak; Vojtěch Rödl — 1996

Acta Arithmetica

On generalized shift graphs

Christian Avart; Tomasz Łuczak; Vojtěch Rödl — 2014

Fundamenta Mathematicae

In 1968 Erdős and Hajnal introduced shift graphs as graphs whose vertices are the k-element subsets of [n] = 1,...,n (or of an infinite cardinal κ ) and with two k-sets $A = a ₁, . . ., a_{k}$ and $B = b ₁, . . ., b_{k}$ joined if $a ₁ < a ₂ = b ₁ < a ₃ = b ₂ < \dots < a_{k} = b_{k - 1} < b_{k}$ . They determined the chromatic number of these graphs. In this paper we extend this definition and study the chromatic number of graphs defined similarly for other types of mutual position with respect to the underlying ordering. As a consequence of our result, we show the existence of a graph with interesting...