Is the (Triangle-Free) Henson Graph Pseudofinite?

• Henson graph $H_3$: the unique (up to isomorphism) countable graph that is homogeneous, triangle-free, and universal for countable triangle-free graphs — every countable triangle-free graph embeds into it.
• Pseudofinite structure: a structure that is elementarily equivalent to an ultraproduct of finite structures (equivalently, every sentence true in it is true in some finite structure of the same signature).

• The analogous question is resolved for the ordinary (triangle-permitting) random graph, which is pseudofinite.
• Various structural and combinatorial properties of $H_3$ relevant to pseudofiniteness (e.g. related to Ramsey-type and homogeneity properties of $K_3$-free graphs) have been studied, but a resolution of pseudofiniteness itself remains open.

This question sits at the intersection of model theory and finite combinatorics/finite model theory, and is a natural test case for understanding which homogeneous structures built by Fraïssé-style constructions are pseudofinite.

• Henson, C. W. "A family of countable homogeneous graphs." Pacific Journal of Mathematics, 1971 (introduces the Henson graphs).
• Various papers on pseudofinite structures and generalized Fraïssé constructions discuss this question; readers should consult recent model theory literature on pseudofiniteness for up-to-date status.