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1
Content available remote

On families of weakly dependent random variables

100%
Łuczak T.
Banach Center Publications
|
2011
|
tom 95
|
nr 1
123-132
EN
Let $𝒢ₙ^{(k)}$ be a family of random independent k-element subsets of [n] = {1,2,...,n} and let $𝓗 (𝒢ₙ^{(k)},ℓ) = 𝓗 ₙ^{(k)}(ℓ)$ denote a family of ℓ-element subsets of [n] such that the event that S belongs to $𝓗 ₙ^{(k)}(ℓ)$ depends only on the edges of $𝒢ₙ^{(k)}$ contained in S. Then, the edges of $𝓗 ₙ^{(k)}(ℓ)$ are 'weakly dependent', say, the events that two given subsets S and T are in $𝓗 ₙ^{(k)}(ℓ)$ 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 some questions which, despite the progress which has been made for the last few years, remain to puzzle researchers who work in the area of probabilistic combinatorics.
2
Content available remote

Small bases for finite groups

64%
Łuczak T., Schoen T.
Colloquium Mathematicum
|
1998
|
tom 78
|
nr 1
35-37
3
Content available remote

On random subsets of projective spaces

64%
Kordecki W., Łuczak T.
Colloquium Mathematicum
|
1991
|
tom 62
|
nr 2
353-356
4
Content available remote

On strongly sum-free subsets of abelian groups

64%
Łuczak T., Schoen T.
Colloquium Mathematicum
|
1996
|
tom 71
|
nr 1
149-151
EN
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 $n ≥ n_0$ 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 $a_{i_j}a_{i_k} ≠ a_m$ for $1 ≤ j < k ≤ l$ and $1 ≤ m ≤ n$. 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.
5
Content available remote

On the maximal density of sum-free sets

64%
Łuczak T., Schoen T.
Acta Arithmetica
|
2000
|
tom 95
|
nr 3
225-229
6
Content available remote

On a problem of Konyagin

64%
Łuczak T., Schoen T.
Acta Arithmetica
|
2008
|
tom 134
|
nr 2
101-109
7
Content available remote

Arithmetic progressions of length three in subsets of a random set

51%
Kohayakawa Y., Łuczak T., Rödl V.
Acta Arithmetica
|
1996
|
tom 75
|
nr 2
133-163
8
Content available remote

On generalized shift graphs

51%
Avart C., Łuczak T., Rödl V.
Fundamenta Mathematicae
|
2014
|
tom 226
|
nr 2
173-199
EN
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 $A = {a₁,...,a_{k}}$ and $B = {b₁,...,b_{k}}$ joined if $a₁ < a₂ = b₁ < a₃ = b₂ < ⋯ < a_k = b_{k-1} < b_k$. 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 disparity of chromatic behavior of finite and infinite subgraphs. For any cardinal κ and integer l, there exists a graph G with |V(G)| = χ(G) = κ but such that, for any finite subgraph F ⊂ G, $χ(F)≤ log_{(l)}|V(F|$, where $log_{(l)}$ is the l-iterated logarithm. This answers a question raised by Erdős, Hajnal and Shelah.
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