# Examples of “folk theorems”

In this post, Justin gives a quote about Raoul Bott that has this line in it:

He talked about 'folk' theorems... theorems everyone knew, but were never written down.

What are some good/interesting examples of these types of theorems?

• I gave a sort of weak example on page 1 of math.uga.edu/~pete/coveringnumbersv2.pdf -- this is a simple linear algebra result that often gets asked as a problem (e.g. on MO!) but is rarely discussed in standard texts. – Pete L. Clark Jul 18 '10 at 23:18
• Any response to this question ceases to be an example! – Qiaochu Yuan Sep 12 '11 at 1:01

The example I first learned was the following: a 2-D TQFT is equivalent to a Frobenius algebra.

This is discussed and stated as a folk theorem by Voronov; later, a careful proof was written up and published by Lowell Abrams. See also the book by Joachim Kock.

In the context of game theory, the term «folk theorem» has a rather specific meaning...

In category theory there is a 'folk' model structure on the category Cat, where the weak equivalences are the equivalences of categories. There is a similar model structure on 2Cat, with weak equivalences being equivalences of 2-categories (weak ones, I presume) The former was not written down for a long time, but the latter was published by Steve Lack. Andre Joyal is not in favour of the name 'folk model structure', and there was discussion on this at the nForum (starting at that comment and continuing). That the existence of this model structure is a 'folk' theorem is a bit of folklore itself, as pointed out by Joyal at this comment.

• It seems now that people are trying to change the name of this to be the 'canonical' model structure on Cat – David White Mar 18 '13 at 16:57
• This is true, and you can see it in the discussion I link to. – David Roberts Mar 20 '13 at 22:59

On Folk Theorems is an old classic from computer science. Although the title suggests it's about folk theorems in general, it's mostly about the theorem which states, roughly, that programs written in imperative programming languages only need one loop.

I was at a queueing theory lecture recently where the lecturer talked about Little's Theorem and Wolff's PASTA theorem as having been around as folk theorems for a long time before they were published with proof.

In Fudenberg's book Game Theory, the following was listed as a folk theorem:

The folk theorem for repeat games assert that if players are sufficiently patient then any feasible, individual rational payoffs can be enforced by an equilibrium. Thus, in the limit of extreme patience, repeated play allows any payoff to be an equilibrium outcome.

There are quite a few examples in additive combinatorics of theorems or tricks that were talked about and 'known' a few years before anyone published a proof of them.

For example, let $\phi(n)$ be the largest number such that every set A of n reals contains a subset B of cardinality $\phi(n)$ such that no element of A can be represented as the sum of two distinct elements of B ('B is sum-free with respect to A').

It was remarked by both Klarner and Erdos that $\phi(n)\geq\log n-O(1)$ for large n, but it was ten years before Choi published a proof of this (a simple application of Turan's theorem on independent sets in graphs).

Presumably phenomena like this occurs because those who think of it see it as too simple or straightforward to be worth the bother of publishing.

A different type of example is the idea that if $f:G\to\mathbb{C}$ is a function on a finite abelian group $G$ with a small $L^2$ norm, then it can be decomposed as the sum of structured parts (with a small error term).

For example, $f=f_1+f_2+f_3$, where $f_1$ is the linear combination of a small number of characters, $f_2$ is Gowers uniform and $f_3$ has $L^2$ norm less than $\epsilon$.

This kind of folk theorem arises because it is a commonly applied heuristic that can be made precise in a variety of different ways, often jury-rigged for a specific application.

Stark, The Gauss class-number problems, available at http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/stark.pdf writes, on page 251, "We define the Epstein zeta functions, $$\zeta(s,Q)=(1/2)\sum_{m,n\ne0,0}Q(m,n)^{-s}$$ ... Theorem 4.1 (Folk Theorem.) Let $c\gt1/4$ be a real number and set $$Q(x,y)=x^2+xy+cy^2,$$ with discriminant $d=1-4c\lt0$. Then for $c\gt41$, $\zeta(s,Q)$ has a zero $s$ with $\sigma\gt1$."

He follows this with a "Folk proof."

My advisor once told me that the following statement (which I read in Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres) was a Folk Theorem:

For $p>2$ and in a certain range, the Adams Spectral Sequence coincides with the homology Bockstein spectral sequence

It turns out the range is $t<(2p-1)s-2$, and a reference is Haynes Miller's paper

There are some "folkish" elements in something I just published:

The "80/20" account of Pareto's law circulates among management people who don't care about mathematics, and the probability density proportional to $x \mapsto x^{-\alpha - 1}$ on $(x_0,\infty)$ for some $x_0>0$ (and 0 elsewhere) is found in many probability and statistics books, and actually gets used in various fields to which mathematics gets applied. But the idea that they are in some sense the same thing seems to have circulated only in a "folk" manner for many years until now.

Another nice type of 'folk theorems' I have seen is of a sort where some relatively straight forward generalization of a well established theorem is assumed and then used for its heuristic or explanatory value. I find this is often used in fields where mathematicians interact with non-mathematicians and although it is completely non-rigorous (and sometimes even misleading!) most often it helps in exposition and for building intuition.

An example would be the "folk theorem of evolutionary game theory" (as used by Hofbauer and Sigmund, BAMS 2003) on certain kinds of correspondences of Nash equilibrium and dynamic approaches.