We discuss some basic geometry of sets definable in the Urysohn sphere using only finitely many parameters and briefly remark on the case of arbitrary definable sets. Then we discuss definable functions in the Urysohn sphere satisfying a special syntactic property.

We investigate several variants of the homomorphism problem: given two relational structures, is there a homomorphism from one to the other? The input structures are possibly infinite, but definable by first-order interpretations in a fixed structure. Their signatures can be either finite or infinite but definable. The homomorphisms can be either arbitrary, or definable with parameters, or defi...

Let T be a (first order complete) dependent theory, C a κ̄-saturated model of T and G a definable subgroup which is abelian. Among subgroups of bounded index which are the union of < κ̄ type definable subsets there is a minimal one, i.e. their intersection has bounded index. See history in [Sh:876]. We then deal with definable groups for 2-dependent theories, a wider class of first order theories...

It is true in the Solovay model that every countable ordinal-definable set of sets of reals contains only ordinal-definable elements.