By Ohkawa's theorem, the Bousfield lattice $B$ (of the $\infty$-category of spectra) is a small, complete lattice with $2^{\aleph_0} \leq |B| \leq 2^{2^{\aleph_0}}$ (the exact cardinality is an open question). This implies the existence of various other cardinal invariants of $B$, which I'm curious about.
Questions:
For each Bousfield class $\langle E \rangle$, there is a representative $E$ of minimal size (measured, say, by number of cells) -- call this "size" of $\langle E \rangle$. Because $B$ is a set, we may ask:
What is the supremum of sizes of Bousfield classes?
The fact that $|B| \leq 2^{2^{\aleph_0}}$ doesn't give any bound on this supremum; it just ensures that it exists. So I'd be interested even in a loose upper bound. Anything that might be known about "gaps in the spectrum" would be interesting too -- i.e. cardinals below the supremum which are nevertheless not the size of any Bousfield class.
Because $B$ is a set, there exists a small set $S$ of spectra which "detects" Bousfield equivalence, in the sense that $\langle E \rangle = \langle F \rangle \Leftrightarrow (\forall X \in S,\, E \wedge X = 0 \Leftrightarrow F \wedge X = 0)$. We can ask
What is the smallest cardinal $\lambda$ such that we can choose $S$ to consist of spectra of size $<\lambda$?
Again, size may be measured by number of cells. It would be particularly interesting if $S$ could be taken to consist of the finite-type spectra, but that can't be known because it would imply $|B| = 2^{\aleph_0}$.
The Erdos-Rado theorem implies that if $|B| > 2^{\aleph_0}$, then $B$ contains either an uncountable antichain, an uncountable well-ordered chain, or an uncountable reverse well-ordered chain. So a natural approach to showing $|B| = 2^{\aleph_0}$ would be to rule out such structures.
Are any such structures known to exist in $B$ (or conversely are any such ruled out)?
Because $B$ is small and complete, it is locally presentable.
For which $\lambda$ is $B$ locally $\lambda$-presentable?
