Supersaturation in posets and applications involving the container method

In addition, we apply our supersaturation results to obtain (a) upper bounds on the number of antichains in these posets and (b) asymptotic bounds on the cardinality of the largest antichain in p-random subsets of these posets which hold with high probability (for p in a certain range). The proofs of these results rely on a 'container-type' lemma for posets which generalises a result of Balogh, Mycroft and Treglown [6]. We also state a number of open problems regarding supersaturation in posets and counting antichains.