Just a couple more notes on this setup so I don’t forget.
First of all, I set this up as a problem about G acting on G/H, i.e. G acting transitively on a set. But the transitivity is actually not critical. The group could be trivial! If we call the set Σ, then G could be the trivial group, and we’d just be asking for large subsets S of Σ such that S^m was contained in some specified subset R of Σ^m. There are perfectly questions like that! I’ve seen this come up for instance when Σ is k and R is an algebraic subvariety of k^m. So maybe you could think of the setup of the blogpost as “questions like that where the problem admits a really big automorphism group…
Just a couple more notes on this setup so I don’t forget.
First of all, I set this up as a problem about G acting on G/H, i.e. G acting transitively on a set. But the transitivity is actually not critical. The group could be trivial! If we call the set Σ, then G could be the trivial group, and we’d just be asking for large subsets S of Σ such that S^m was contained in some specified subset R of Σ^m. There are perfectly questions like that! I’ve seen this come up for instance when Σ is k and R is an algebraic subvariety of k^m. So maybe you could think of the setup of the blogpost as “questions like that where the problem admits a really big automorphism group, maybe so big that the elements of Σ are homogeneous for it.” I do think the transitivity cuts you down to a particularly congenial and natural class of such problems; “problems with no moduli” if you like.
Yet one more note. I think one could work with more general tensor categories than the category of sets. For instance, one could take V to be a representation of G, and let R be a subrepresentation of V^{⊗m}, and ask: how large can a subspace (not a subrepresentation) W of V be with the property that W^{⊗m} lies in R? And tensor powers are actually not necessarily the most natural thing to do to V; probably rather than choosing m and tensoring m times, we should choose a partition λ and apply the λ Schur functor. I assumed that once again lots of standard problems would fall out of this setup, but to be honest, I couldn’t think of any. Can you?
I guess here’s one baby case. Let V be the standard permutation representation of S_n. Then Sym^d V is homogeneous forms in n variables with S_n permuting coordinates. Let R be the subrepresentation of forms F such that the coefficients of X_i^d sum to 0. Then a subspace W of linear forms such that Sym^d W lies in R is just (I think) a linear subspace of the diagonal hypersurface sum x_i^d = 0 in P^{n-1}, and such a subspace has dimension at most n/2 or so I think (pair up coordinates to make x_i^d = – x_{i+1}^d; maybe you have to be a little more clever if d is even and -1 doesn’t have a square root?)
Tagged combinatorics, representation theory