-color partitions with weighted differences equal to minus two.
Let denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, . They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for . Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for . Furthermore, they also constructed infinite families of congruences for modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several new infinite...
A slight modification of the proof of Szemerédi’s cube lemma gives that if a set satisfies , then must contain a non-degenerate Hilbert cube of dimension . In this paper we prove that in a random set determined by for , the maximal dimension of non-degenerate Hilbert cubes is a.e. nearly and determine the threshold function for a non-degenerate -cube.