Assuming AD + (V = L(R)), it is shown that for κ an admissible Suslin cardinal, o(κ) (= the order type of the stationary subsets of κ) is "essentially" regular and closed under ultrapowers in a manner to be made precise. In particular, o(κ) ≫ κ +, κ ++ , etc. It is conjectured that this characterizes admissibility for L(R).

We formulate and prove a combinatorial property assuming AD + V = L(R). As a consequence, we show that every regular κ which is either a Suslin cardinal or the successor of a Suslin cardinal is δ 2 1 -supercompact. In particular, all the projective ordinals δ 1 n are δ 2 1 -supercompact.

We survey some results and problems arising from a classic problem of Steinhaus: Is there a subset S of R 2 such that each isometric copy of mathbbZ 2 (the lattice points in the plane) meets S in exactly one point.

We show that all the projective ordinals $\delta^1_n$ are supercompact through their supremum $\aleph_{\varepsilon 0}$, and a ways beyond.

We work under the assumption of the Axiom of Determinacy and associate a measure to each cardinal $\kappa < \aleph_{\varepsilon_0}$ in a recursive definition of a canonical measure assignment. We give algorithmic applications of the existence of such a canonical measure assignment (computation of cofinalities, computation of the Kleinberg sequences associated to the normal ultrafilters on all projective ordinals).

Let E be a coanalytic equivalence relation on a Polish space X and (A n ) n∈ω a sequence of analytic subsets of X. We prove that if lim sup n∈K A n meets uncountably many E-equivalence classes for every K ∈ [ω] ω , then there exists a K ∈ [ω] ω such that ⋂ n∈K A n contains a perfect set of pairwise E-inequivalent elements.

Let X and Y be uncountable Polish spaces. We show in ZF that there is a coanalytic subset P of X × Y with countable sections which cannot be expressed as the union of countably many partial coanalytic, or even PCA = Σ 1 2 , graphs. If X = Y = ω ω , P may be taken to be Π 1 1 . Assuming stronger set theoretic axioms, we identify the least pointclass such that any such coanalytic P (...) can be expressed as the union of countably many graphs in this pointclass. This last result is extended (under suitable hypotheses) to all levels of the projective hierarchy. (shrink)