Asymptotic enumeration, a cornerstone of enumerative combinatorics, focuses on determining the number of discrete structures as their size approaches infinity. In the theory of partially ordered sets (posets), this area is crucial for understanding the statistical and structural properties of large random posets, serving as a foundational tool for various applications, including the study of zero-one laws in finite model theory [4] [3].

This talk will give an overview of asymptotic enumeration for posets. We will review some existing results on the problem [1] [2], and explore some novel techniques being used to derive asymptotic formulas for the number of posets of a given size. By examining the behaviour of large random posets, we gain insights into the emergence of typical structures and the properties they almost surely possess. This presentation will highlight recent progress and open questions, demonstrating the impact of these enumeration techniques on diverse fields, from pure mathematics to theoretical computer science.

1. Daniel Kleitman and Bruce Lee Rothschild, Asymptotic enumeration of partial orders on a finite set, Transactions of the American Mathematical Society, vol.205 (1975), no.1, pp.205–220.
2. Deepak Dhar, Asymptotic enumeration of partially ordered sets, Pacific Journal of Mathematics, vol.90 (1980), no.2, pp.299–305.
3. Joseph Y. Halpern and Bruce Kapron, Zero-one laws for modal logic, Annals of Pure and Applied Logic, vol.69 (1994), no.2, pp.157–193.
4. Rineke Verbrugge, Zero-one laws with respect to models of provability logic and two Grzegorczyk logics, Advances in Modal Logic, (2018), pp.115–120.