I don’t think our result quite shows that, since we only forbid $5$- and $6$-tuples of nonbases in a specific structure; potentially we still keep rank-$4$ minors on eight elements with $> 4$ nonbases that are not Vamos-like.

However, I’ve thought about similar questions, and I think it should be the case that there is some constant $\alpha > 0$ so that there are a lot of matroids whose $m$-element minors all have at most $\alpha m$ nonbases; for $s \approx m/2$ this is very few indeed.

In fact, I think we can show, by sampling from the Graham/Sloane construction instead of the whole Johnson graph, that for a worse value of $\alpha$, the same statement holds when $n^{-2}$ is replaced with $n^{-h}$ for any $h$ strictly greater than $1$.

This is certainly eroding my belief in the conjecture that for sparse paving $N$, almost every matroid contains $N$ as a minor.

]]>The proof of your final claim shows that there are very many sp matroids (at least $c n^{-2}\binom{n}{n/2}$ in the exponent) so that each minor on $m=8$ elements of rank $s=4$ has at most $k=4$ nonbases. What function $k=k(m,s)$ makes this true in general?

(Point 2 just above that claim seems to have swallowed some surrounding text.)

]]>https://en.wikipedia.org/wiki/Dickson%27s_lemma

and

https://en.wikipedia.org/wiki/Buchberger%27s_algorithm

Dicksons lemma follows by a straightforward induction argument.

So no need to apply Zorns Lemma here.

]]>I think your interpretation of the ultraproduct is exactly right.

]]>But these other proofs (especially 2) are really elegant.

Proof #3 — which I don’t really understand — looks a lot like ‘the product partial field of infinitely many finite fields has a homomorphism to a field of char 0’.

]]>I think there may be an opportunity for some reverse mathematics here: which axiom scheme is necessary to prove Theorem 1?

]]>I was a bit confused by proof #2 at first, but I think you made a typo. Don’t you mean that if M has no representation in char. 0, then T + { *not* S_p : p prime } is inconsistent?

]]>that sounds like a really good idea. As you suggest, the algebraic cycle matroid looks like it is coming from the singleton partition in this way. It would be great to hear if you make any progress with this idea. For example, which bipartitions of the set of all ends induce infinite matroids in this way? What do you think?

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