The Exact Saturation Number for the Diamond

TL;DR

This paper determines the exact minimal size of a family of subsets of [n] that avoids an induced diamond poset but becomes one upon adding any set, establishing it as exactly n+1.

Contribution

It proves that the minimal family size avoiding an induced diamond is exactly n+1, resolving a longstanding open problem in extremal set theory.

Findings

The minimal size of such a family is exactly n+1.

A maximal chain achieves this minimal size.

The result confirms the lower bound matches the upper bound for all n.

Abstract

What is the smallest size of a family of subsets of $[n]$ such that it does not contain an induced copy of $Q_2$ as a poset (known as the \textit{diamond}), but adding a new set creates such a copy? It is easy to see that a maximal chain has this property, and thus the answer is at most $n+1$. Despite the simplicity of the diamond structure, the lower bound stagnated at $\sqrt n$ for quite some time, until recently the authors obtained a linear lower bound. In this paper, we fully solve this question showing that such a family must have size at least $n+1$.