On families of weakly dependent random variables

• Autorzy: Tomasz Łuczak
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 05C80: Random graphs
• 05C35: Extremal problems
• 11B25: Arithmetic progressions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2011
• Tom: 95
• Numer: 1
• Strony: 123-132

Daty

wydano

2011

Twórcy

autor

Tomasz Łuczak

• Adam Mickiewicz University, Faculty of Mathematics and Computer Science, 61-614 Poznań, Poland

Identyfikatory

DOI

10.4064/bc95-0-9