Maximal Arithmetic Progressions in Random Subsets

Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0, 1}N . By an application of the Chen-Stein method, we show that U −2 log N/ log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W (N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U / log N converges almost surely to 2/ log 2, while W / log N does not converge almost surely (and in particular, lim sup W / log N ≥ 3/ log 2).