I want to use this space to sketch the proof of the following known result:
Proposition: If is Jónsson for singular, then there is cofinal in with such that carries a better scale.
All one needs to do here is show that and then use that to construct a better scale. So, the Jónsson assumption is moderately silly.
Proof: Let . We begin by showing that . This is pretty standard, so fix a scale with and we will show that co-boundedly many elements of must be Jónsson. Otherwise, thin out so that every element carries a Jónsson cardinal. Let be large enough regular and let be such that , , and .
It suffices to show that for each sufficiently large . So fix , and note that we can find such that . Otherwise, for all large enough and so . But then, there is some with such that . But this is silly, as and so there is some with , but as it’s definable from parameters. So now we can find with and , and such that carries a Jónsson algebra. But then, . As was arbitrary, we get a contradiction.
So now we let be such that , and every carries a Jónsson cardinal. Then, , otherwise we could find a generator for . In particular, (since ) and so , which is absurd by our earlier argument. Thus, we have that and in particular we get that is -directed. With this in hand, the rest of the proof is pretty easy.
We begin by fixing a silly square sequence . We inductively construct a sequence as follows:
Stage : Just let be any function in .
Stage : Here we let be any pointwise larger function in .
Stage limit: Now we have to do work. For each , let if , and otherwise. Note that these functions are non-zero almost everywhere. Then we let be such that for each (here is where we use the directedness as ).
Note that has an exact upper bound by construction with the property that is bounded for each . First we claim that is better in . In other words, we need that:
for every of the appropriate cofinality, there is a club such that for every in , there is a such that in implies that for every in .
This follows directly from the construction, so let be of cofinality above . Then let be club, and let . Note then that so let be such that for all . In particular, we see that . So we have betterness.
Then the usual argument shows that (by thinning out if necessary) carries a better scale, which is what we actually want.