# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Thursday, November 24, 2016, 12:15 pm

**Duration**: 30 minutes

**Location**: CAB G51

**Speaker**: Jonathan Noel

## Supersaturation in Posets and Applications Involving the Container Method

We consider 'supersaturation' problems in partially ordered sets (posets) of the following form. Given a finite poset $P$ and an integer $m$ greater than the cardinality of the largest antichain in $P$, what is the minimum number of comparable pairs in a subset of $P$ of cardinality $m$? We provide a framework for obtaining lower bounds on this quantity based on counting comparable pairs relative to a random chain and apply this framework to obtain supersaturation results for three classical posets: the boolean lattice, the collection of subspaces of $\mathbb{F}_q^n$ ordered by set inclusion and the set of divisors of the square of a square-free integer under the 'divides' relation. The bound that we obtain for the boolean lattice can be viewed as an approximate version of a known theorem of Kleitman. 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 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. Based on joint work with Alex Scott and Benny Sudakov.

Upcoming talks | All previous talks | Talks by speaker | Upcoming talks in iCal format (beta version!)

Previous talks by year: 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M | admin login