Turán and Ramsey problems for alternating multilinear maps

Turán and Ramsey problems for alternating multilinear maps, Discrete Analysis 2023:12, 22 pp.

Ramsey’s theorem (in its finite version) states that for every positive integer $k$ there exists a positive integer $n$ such that every graph with $n$ vertices contains a clique or an independent set with $k$ vertices. This result is just the beginning of what has become a huge field. Typically, a result in Ramsey theory takes the following form: one is given a structure $S$ and a family $\mathcal S$ of structured subsets of $S$, and the result says that however one colours the elements of $S$ with at most $r$ colours, there must be a substructure $S'\in\mathcal S$ with all its elements of the same colour. For example, it follows from the Hales-Jewett theorem that for every prime $p$ and every pair of positive integers $r$ and $k$ there exists a positive integer $n$ such that however the points of $\mathbb F_p^n$ are coloured with $r$ colours, there must be an affine subspace of $\mathbb F_p^n$ of dimension at least $k$ with all its points of the same colour.

Ramsey theory is not just a collection of attractive theorems, since Ramsey-theoretic results often have surprising applications. Typically, these will tell us that a set must have a subset with one of two very different properties. A simple example of this is the statement that every sequence of real numbers has a subsequence that is either increasing or decreasing (though that can be proved directly with a simpler argument). There are also more complicated examples where one obtains either a highly structured subset or a subset that is “maximally unstructured” in some suitable sense.

Some results that follow this general pattern are not directly deduced from classic results of Ramsey theory; they can therefore be regarded as a kind of extension of Ramsey theory. One such result, a “quantum Ramsey theorem” relevant to this paper, was proved by Nik Weaver in 2016. Define a quantum system to be a linear subspace of $M_n(\mathbb C)$ that contains the identity and is closed under taking adjoints. Given a quantum system $\Lambda$ and a subspace $V\subset\mathbb C^n$, define the restriction of $\Lambda$ to $V$ to be the family $\{PAP:A\in\Lambda\}$, where $P$ is the orthogonal projection from $\mathbb C$ to $V$. Weaver proved that for every $s$ there exists $n$ (the bound obtained is $s^3-s+1$) such that for every quantum family $\Lambda\subset M_n(\mathbb C)$ there is a subspace $V$ of dimension $s$ such that the restriction of $\Lambda$ to $V$ is either as large as possible – that is, it is isomorphic to $M_s(V)$ – or as small as possible – that is, it consists of multiples of $P$ (which from $V$’s perspective look like multiples of the identity).

An interesting example of Weaver’s theorem is where $\Lambda$ consists of all diagonal matrices. It is not hard to show that the restriction of $\Lambda$ even to a 2-dimensional subspace must have dimension greater than 1, so, somewhat surprisingly, the theorem yields a subspace $V$ where the restriction is isomorphic to $M_n(V)$. (This becomes less surprising when one notes that if $D$ and $D'$ are diagonal matrices and $P$ is an orthogonal projection, it does not necessarily follow that $PDP$ and $PD'P$ commute.)

The main result of this paper sounds quite similar to Weaver’s theorem, but there are important differences that cause it to need a significantly different proof. It concerns alternating bilinear forms. If $U$ is a vector space over a field $\mathbb F$, then an alternating form on $U$ is a bilinear map $f:U\times U\to\mathbb F$ such that $f(u,u)=0$ for every $u\in U$ (which implies that $f(u,v)=-f(u,v)$ for every $u,v\in U$, and conversely if $\mathbb F$ does not have characteristic 2). Write $\Lambda(U)$ for the space of all alternating bilinear forms on $U$. Then the main theorem of this paper states that if $U$ is of high enough dimension, then for every subspace $\mathcal A$ of $\Lambda(U)$, either there is an $s$-dimensional subspace $V$ of $U$ such that the restriction of $\mathcal A$ to $V$ contains just the zero map, or there is a $t$-dimensional subspace $W$ such that the restriction of $\mathcal A$ to $W$ is $\Lambda(W)$.

The bound obtained for the dimension of $U$ is $st^4$. Since this is a power-type bound, it is immediately clear that this result is not a simple consequence of Ramsey’s theorem. The important differences between this and Weaver’s result are that it works for arbitrary fields, and that diagonal matrices, which play a key role in Weaver’s arguments, are absent here.

An indication that this is a natural direction to pursue, and not just a problem formulated for the sake of it, is that it can be reformulated as a statement about $p$-groups. It says that among $p$-groups that are nilpotent of class 2 (that is, the commutators $xyx^{-1}y^{-1}$ commute with all other elements) every sufficiently large group $G$ has a subgroup $S$ with one of the following two properties.

1. $S[G,G]/[G,G]$ is isomorphic to $\mathbb F_p^s$.
2. $S$ is isomorphic to the free nilpotent $p$-group of class 2 with $t$ generators.

The free nilpotent $p$-group of class 2 on $t$ generators $x_1,\dots,x_t$ is the group with all relations of the form $u^t=e$ and $u[v,w]=[v,w]u$, where $u,v,w$ are arbitrary words.

Very roughly, this says that $G$ has a subgroup that is either as structured as possible (modulo the commutator subgroup $[G,G]$ it is Abelian) or as unstructured as possible (it is free, up to the constraints that it is contained in a $p$-group that is nilpotent of class 2).

The paper contains a discussion of several other interesting connections with different parts of mathematics, as well as some natural conjectures, which include in particular a conjecture that would extend the main result of this paper from bilinear to multilinear forms. Thus, as well as proving Ramsey-theoretic results with attractive and non-obvious statements, it is potentially opening up a new subfield of Ramsey theory.

This talk by the author contains more information about the background to these results.