### Recent developments in poset saturation - general bounds, and the diamond

Combinatorics Seminar

20th February 2024, 11:00 am – 12:00 pm

Fry Building, 2.04

A poset is short for a partially ordered set. The most common example of a poset is the power set of [n] with the partial relation given by inclusion. Given a fixed poset P, we say that a family F of subsets of [n] is P-free if there is no induced copy of P formed by elements of F. We further say that F is P-saturated if it is P-free and, for any other set X not in F, the family formed by adding X to F contains an induced copy of P. The size of the smallest P-saturated family is called the induced saturation number of P.

The natural question is: what can we say about the saturation number? Even for simple posets such as the the antichain and the butterfly, the question has proved difficult – for the diamond poset the question is surprisingly wide open.

How about the saturation number for an arbitrary poset P? Freschi, Piga, Sharifzadeh and Treglown proved that the saturation number for any poset is either bounded, or at least \sqrt n. What about the upper bound?

In this talk we will discuss recent developments for the diamond question, as well as new bounds for an arbitrary poset.

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