Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a *combinatorial Hilbert space*, namely the set of all supports of all vectors in $V$.

My question: Can one give a combinatorial characterization of combinatorial Hilbert spaces?

To start things off, I'll mention obvious (or at least nearly obvious) necessary conditions on an ${\cal S}_V$:

1) Closure under countable (and thus here arbitrary) unions [*countable* anticipates generalization to larger ordinals than $\omega$];

2) For sets $A_1,\ldots,A_n\in {\cal S}_V$ with none contained in the union of all the others, and at least $n-1$ integers belonging to at least two $(A_i)$'s, there exists non-empty $\ C\subsetneq \bigcup A_i$ with $|C|\geq n-1$ such that $\bigcup A_i\setminus C\in {\cal S}_V$;

3) For $A,B \in {\cal S}_V$ with $\emptyset \not= B \subsetneq A$, there exists non-empty $C\subset B$ with $A\setminus C \in {\cal S}_V$.

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