This will be an invited talk for the AMS-ASL special session on Surreal Numbers at the 2016 Joint Mathematics Meetings in Seattle, Washington, January 6-9, 2016.

**Abstract.** The hypnagogic digraph, a proper-class analogue of the countable random $\mathbb{Q}$-graded digraph, is a surreal-numbers-graded acyclic digraph exhibiting the set-pattern property (a form of existential-closure), making it set-homogeneous and universal for all class acyclic digraphs. A natural copy of this canonical structure arises during the course of the usual construction of the surreal number line, using as vertices the surreal-number numerals $\{\ A \mid B\ \}$. I shall explain the construction and elementary theory of the hypnagogic digraph and describe recent uses of it in connection with embeddings of the set-theoretic universe, such as in the proof that the countable models of set theory are linearly pre-ordered by embeddability.

Slides | schedule | related article | surreal numbers (Wikipedia)

The finite pattern property for graded digraphs generalizes a corresponding property characterizing the countable random graphs, and asserts that for any disjoint finite sets of vertices A, B and C, with a grading level $\alpha$ such that every node in A is below $\alpha$ and every node in B is above $\alpha$, then there is a node v on level $\alpha$ such that $a\rightharpoonup v\rightharpoonup b$ for all $a\in A$ and $b\in B$, and with no edges between v and any node in $C$. The set pattern property property for the hypnagogic digraph makes this assertion for all sets A, B and C, regardless of size.