Title: The induced saturation problem for posets
Speaker: Andrea Freschi
Abstract. Given posets P and Q, we say that P is an induced subposet of Q if there exists a
function ϕ : P → Q such that A ⊆ B if and only if ϕ(A) ⊆ ϕ(B), for every A, B ∈ P . Otherwise,
Q is said to be induced P -free. A family F ⊆ 2[n] is induced P -saturated if F is induced P -free and,
for every subset S ∈ 2[n] \ F, P is an induced subposet of F ∪ {S}. The size of the smallest such
family F is denoted by sat∗(n, P ). We show that either sat∗(n, P ) = O(1) or sat∗(n, P ) ≥ 2√n − 2,
improving a previous result by Keszegh, Lemons, Martin, Palvolgyi and Patkos.
It remains open as to whether our result is essentially best possible or not. Keszegh, Lemons,
Martin, Palvolgyi and Patkos conjectured that sat∗(n, P ) ≥ n + 1 in the unbounded case, but even
for the diamond poset (♢) this is not known. We also discuss a number of related open problems
in the area.
This is based on joint work with Simon Piga, Maryam Sharifzadeh and Andrew Treglown.
