A search for theorems which appear to have very few, if any hypotheses

Question

I'm interested in theorems which appear to have very few, if any hypotheses. Essentially a search for unexpected regularity or pattern in a relatively unstructured situation.

By "few hypotheses" I mean theorems which start "take any triangle", or "take any three circles". Similarly, the conclusion of the theorem ought to be really surprising. I know this is a little vague, but I've deliberately left it that way.

Perhaps my favourite here is Morley's theorem. This applies to any triangle, but has a very surprising conclusion. Contrast this with Pythagoras' theorem (needs a right angled triangle: too special!) or Viviani's Theorem (needs an equilateral triangle: too special!).

Can you help me gather a collection? I've made a very preliminary start here: http://tube.geogebra.org/book/title/id/2673817

Part of my underlying interest is in the aesthetic, and what professional mathematicians think is "significant", "surprising" or when exceptional cases mean the "take any .... except ..." means the theorem isn't so general after all.

Please don't be shy. I'd love to know what your favourites are. They don't have to be in geometry either.....

Chris Sangwin

Answer 1

Theorem. Every group has a terminating transfinite automorphism tower.

Start with any group $G$, compute $\text{Aut}(G)$ and $\text{Aut}(\text{Aut}(G))$ and so on, iterating transfinitely, mapping each to the next via inner automorphisms and taking direct limits at limit stages. Eventually, one arrives at a fixed point, a group that is isomorphic to its automorphism group by the natural map.

• Joel David Hamkins, Every group has a terminating transfinite automorphism tower, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3223--3226.

Answer 2

Every natural number can be written as the sum of four integer squares. (As opposed to the case of three integer squares, which requires certain hypothesis.)

Answer 3

The theorem of Nash and Tognoli says that any compact smooth manifold is diffeomorphic to a nonsingular real algebraic set.

Answer 4

1. There are infinitely many prime numbers.

2. Every integer is a product of primes, in essentially unique way.

   (Theorems with NO hypotheses:-)

Answer 5

Any linear bounded operator on a Hilbert space can be written as linear combination of four unitary operators.

Answer 6

Does the fundamental theorem of algebra count?

Take any (non-constant) polynomial. Then it has a zero among the complex numbers. I suppose this was quite surprising once complex numbers were new.

Answer 7

The Feit-Thompson theorem. Every group of odd order is solvable.

Answer 8

Back in 1907-08, William Henry Young (possibly in part, joint work with his wife, Grace Chisholm Young) proved several "nice behavior" results for arbitrary real-valued functions of a real variable.

In Theorem 6 (p. 82) of [1], Young shows that for co-countably many real numbers $c$ we have

$$\liminf_{x \rightarrow c^{-}}f(x) \; = \; \liminf_{x \rightarrow c^{+}}f(x) \; \leq \; f(c) \; \leq \; \limsup_{x \rightarrow c^{-}}f(x) \; = \; \limsup_{x \rightarrow c^{+}}f(x)$$

(Note: There is a typo in the 2nd inequality at the top of p. 82: $f < {\phi}_R$ should be $f > {\phi}_{R}.)$

This implies two countability results that are now well known. The first is that for an arbitrary function, if both the left limit and the right limit exist at each point, then these unilateral limits can disagree for at most countably many points. The second is that a function can have at most countably many removable discontinuities. Incidentally, this second result was rediscovered by the Romanian mathematician Alexandru Froda in the late 1920s, and there is currently a Wikipedia page titled Froda's Theorem that is misleading at best (see Brian S. Thomson's comments here).

At the 1908 International Congress of Mathematicians, Young announced (see the bottom of p. 54 of [2]) that for co-countably many real numbers $c,$ the left cluster set of $f$ at $c$ is equal to the right cluster set of $f$ at $c.$

This result was proved in [3], where Young additionally showed that at each point of the co-countable set the value of the function belongs to the cluster set. Thus, for co-countably many real numbers $c$ we have

$$C^{-}(f,c) \; = \; C^{+}(f,c) \;\; \text{and} \;\; f(c) \in C^{+}(f,c)$$

This is a seemingly much stronger result than Young's 1907 result, since the 1907 result simply says that the endpoints of the unilateral cluster sets can only differ at countably many points, without saying anything about the distribution of the points belonging to these cluster sets.

Definition: Given a function $f: {\mathbb R} \rightarrow {\mathbb R}$ and $c \in {\mathbb R}$, we let $C^{+}(f,c)$ be the set of all extended real numbers $y$ (i.e. $y$ can be $-\infty$ or $+\infty$) such that there exists a sequence $\left\{x_{k}\right\}$ with each $x_k > c$ and $x_{k} \rightarrow c$ and $f(x_k) \rightarrow y.$ In other words, $C^{+}(f,c)$ is the set of all numbers (including $-\infty$ and $+\infty$) that can be obtained as a limit of $f$-values when using some sequence converging to $c$ from the right. The left version, $C^{-}(f,c),$ is defined analogously.

Regarding these results, see also §6 on pp. 344-346 of [4].

Another result for arbitrary real-valued functions of a real variable (it has been extensively generalized in various directions, as a google search will show) was published by Henry Blumberg in 1922, and a discussion of it can be found at the following math overflow question: Every real function has a dense set on which its restriction is continuous.

[1] William Henry Young, On the distinction of right and left at points of discontinuity, Quarterly Journal of Pure and Applied Mathematics 39 (1908), 67-83. [Paper dated June 1907.]

[2] William Henry Young, On some applications of semi-continuous functions, Atti del IV Congresso Internazionale dei Matematici [4th International Congress of Mathematicians] (Rome), Volume 2, 49-60. [Published version of talk given on 8 April 1908.]

[3] William Henry Young, Sulle due funzioni a più valori costituite dai limiti d'una variabile reale a destra e a sinistra di ciascun pun [On the two functions of multiple values that are determined by the left and right limits of a real variable at each point], Atti della Accademia Reale dei Lincei. Rendiconti. Classe di Scienze fisiche, Matematiche e Naturali (5) 17 #9 (1st semestre) (1908), 582-587. [Paper given at session dated 3 May 1908.]

[4] Andrew Michael Bruckner and Brian Sheriff Thomson, Real variable contributions of G. C. Young and W. H. Young, Expositiones Mathematicae 19 #4 (2001), 337-358.

Answer 9

Liouville's theorem: An entire function is either constant or unbounded.

Similarly, there is little Picard's theorem: The image of an entire function is either $\Bbb C$, $\Bbb C$ without a single point or a single point.

Answer 10

The graph minor theorem. In every infinite sequence of finite graphs, one is a minor of another.

I think this one is a very good match to the original request for "a search for unexpected regularity or pattern in a relatively unstructured situation." An arbitrary finite graph is one of the most "unstructured" mathematical objects possible, and the graph minor theorem asserts the existence of a highly unexpected regularity in this unstructured setting. In many of the examples (mentioned in other answers) involving very little structure, the proof of the theorem is relatively short, confirming one's expectation that if you don't assume much, then there's not much scope for nontrivial logical consequences. But the proof of the graph minor theorem is extraordinarily complicated and deep.

Answer 11

Let G be any finite group. Then the number of conjugacy classes of G is equal to the number of complex irreducible representations of G.

Answer 12

For every holomorphic map from the complex plane to the Riemann sphere, and every $R<\arccos(1/3)$ there exists a disk of radius $R$ in the image in which an inverse holomorphic branch exists. (The constant is best possible).

Answer 13

Among my favorites, Monsky's theorem: it is not possible to partition a square into an odd number of equal-area triangles.

Answer 14

Dirichlet's Theorem on Diophantine Approximation: For every real irrational $\alpha$ there are infinitely many rationals $p/q$ with $$\left|\alpha-{p\over q}\right|<{1\over q^2}$$

Answer 15

Every (smooth) cubic surface in $\mathbb P^3$ (over an algebraically closed field) contains exactly 27 lines.

and, of course (surely this one needs to top any list of this sort :))

If a, b, c, and n are positive integers with $a^n + b^n = c^n$, then n = 1 or 2.

Answer 16

Bertrand's posutlate states that there exists a prime $p$ such that $n<p<2n$ for all $n\in\mathbb{N}$.

In a similar vein, Rosser's theorem gives the bound $p_n\geq n\log n$ for all $n\in\mathbb{N}$, where $p_n$ denotes the $n$'th prime.

Answer 17

Greene's theorem

Curtis Greene, Some partitions associated with a partially ordered set, Journal of Combinatorial Theory Series A, Vol.20(1) (1976) pp 69–79, doi:10.1016/0097-3165(76)90078-9

and the Greene-Kleitman theorem

Curtis Greene, Daniel J Kleitman, The structure of sperner $k$-families, Journal of Combinatorial Theory Series A, Vol.20(1) (1976) pp 41–68, doi:10.1016/0097-3165(76)90077-7

are remarkably deep theorems that hold for any finite partially ordered set.

Greene's theorem. Let $P$ be an $n$-element poset. Let $\lambda_1+\cdots+\lambda_k$ be the largest size of a union of $k$ chains of $P$. Let $\mu_1+\cdots+\mu_k$ be the largest size of a union of $k$ antichains. Let $\lambda=(\lambda_1,\lambda_2,\dots)$ and $\mu=(\mu_1,\mu_2,\dots)$. Then $\lambda$ and $\mu$ are conjugate partitions, i.e., they are weakly decreasing, and the Young diagram of $\mu$ is the transpose of that of $\lambda$.

To see the subtlety of this result, there is for instance a nine-element poset with $\lambda=(5,3,1)$, but $P$ is not a union of a 5-element chain and a 3-element chain.

The fact that $\mu_1$ is the number of parts of $\lambda$ is Dilworth's theorem: the size of the largest antichain of $P$ is equal to the least number of chains whose union is $P$.

Answer 18

Every vector space (or module over a division ring) admits a basis (this includes the empty set as a basis for the zero module). Implicit is the axiom of choice, which I don't consider an extra hypothesis.

All bases of a vector space have the same cardinality.

Answer 19

You may have a look at the book

"Combinatorial Geometry in the Plane" by Hadwiger, DeBrunner, and Klee.

For example, one of my favorites is Proposition #9:

"If an infinite set of points is such that all points are at integral distances from each other, then all of the points lie on a straight line."

I guess one could argue that there are significant hypotheses here, but the conclusion seems surprisingly strong.

Answer 20

Every bounded analytic function in the unit disk has radial limits almost everywhere.

Answer 21

Even more elementary than Morley's theorem is Napoleon's.

Take any triangle, and construct an equilateral triangle on each of its sides. Then their midpoints form an equilateral triangle, too.

I don't know a striking application of the theorem itself. But from the proof, you can also conclude that if the point $D$ inside the triangle $\Delta ABC$ minimises $d(A,\cdot)+d(B,\cdot)+d(C,\cdot)$, then the line segments $AD$, $BD$ and $CD$ meet at angles $\frac{2\pi}3$ (However, here you need an extra assumption that all angles of $\Delta ABC$ are less than $\frac{2\pi}3$, so this does not count for this question).

Answer 22

Here is a theorem that Heinrich Freistuhler & I proved in 1998. There is essentially no assumption.

Let $\phi:{\mathbb R}\rightarrow{\mathbb R}$ be a heteroclinic solution of $\phi'=f(\phi)-q$, where $q$ is some constant. By heteroclinic, we mean that the limits $u_\pm=\phi(\pm\infty)$ exist and are finite. Such functions are viscous standing shocks, that is time-independent solutions of the convection-diffusion equation $$(1)\qquad\partial_tu+\partial_xf(u)=\partial_{xx}^2u.$$ Consider now a function $u_0\in\phi+L^1({\mathbb R})$. Let us define $$h:=\int_{\mathbb R}(u_0-\phi)\,dx.$$ Then the (unique) solution $(x,t)$ of (1) with initial data $u_0$ satisfies (unconditional stability of $\phi$) $$\lim_{t\rightarrow+\infty}\|u(\cdot,t)-\phi(\cdot-h)\|_1=0.$$

This statement assumes neither genuine nonlinearity ($f''$ may vanish arbitrarily), nor decay of $u_0-\phi$ at infinity. The drawback is that the convergence can be arbitrarily slow as $t\rightarrow+\infty$.

Answer 23

Every compact Riemann surface arises from an algebraic plane curve.