## A Note on the Length of Maximal Arithmetic Progressions in Random Subsets

**Authors:** Yilun Shang

Let $U^{(n)}$ denote the maximal length arithmetic
progression in a non-uniform random subset of $\{0,1\}^n$, where $1$
appears with probability $p_n$. By using dependency graph and
Stein-Chen method, we show that $U^{(n)}-c_n\ln n$ converges in law
to an extreme type distribution with $\ln p_n=-2/c_n$. Similar
result holds for $W^{(n)}$, the maximal length aperiodic arithmetic
progression (mod $n$).

**Comments:** 6 Pages.

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### Submission history

[v1] 2014-10-04 20:57:05

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