* Hales-Jewett theorem: For any given n and c, there is a number H such that if the cells of a H-dimensional n × n × n ×...× n cube are coloured with c colours, there must be one row, column, etc.

Notable examples include Szemerédi's theorem, which is such a strengthening of van der Waerden's theorem, and the density version of Hales-Jewett theorem.

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

This implies, by the Bolzano – Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set.

The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.

This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.

The three classes are groups of GF ( 2 ) type ( classified mainly by Timmesfeld ), groups of " standard type " for some odd prime ( classified by the Gilman – Griess theorem and work by several others ), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.

For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:

In fact, Cantor's method of proof of this theorem implies the existence of an " infinity of infinities ".

Together with soundness ( whose verification is easy ), this theorem implies that a formula is logically valid if and only if it is the conclusion of a formal deduction.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

The Paley – Wiener theorem immediately implies that if f is a nonzero distribution of compact support ( these include functions of compact support ), then its Fourier transform is never compactly supported.

Shannon's theorem also implies that no lossless compression scheme can compress all messages.

Bell's theorem implies, and it has been proven mathematically, that compatible measurements cannot show Bell-like correlations, and thus entanglement is a fundamentally non-classical phenomenon.

One might expect that by the Hahn-Banach theorem for bounded linear functionals, every bounded linear functional on C < sub > c </ sub >( X ) extends in exactly one way to a bounded linear functional on C < sub > 0 </ sub >( X ), the latter being the closure of C < sub > c </ sub >( X ) in the supremum norm, and that for this reason the first statement implies the second.

The Heine – Borel theorem implies that a Euclidean n-sphere is compact.

The integrability condition and Stokes ' theorem implies that the value of the line integral connecting two points is independent of the path.

If we choose the volume to be a ball of radius a around the source point, then Gauss ' divergence theorem implies that

Cox's theorem implies that any plausibility model that meets the

The Arzelà – Ascoli theorem implies that if is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence.

For instance, Bell's theorem implies that quantum mechanics cannot satisfy both local realism and counterfactual definiteness.

Noether's theorem implies that there is a conserved current associated with translations through space and time.

However, if the errors are not normally distributed, a central limit theorem often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large.

In fact these uses of AC are essential: in 1950 Kelley proved that Tychonoff's theorem implies the axiom of choice.

To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty.

Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

* Van Kampen's theorem

* Van Kampen's theorem

# Kevin S. Van Horn, " Constructing a logic of plausible inference: a guide to Cox ’ s theorem ", International Journal of Approximate Reasoning, Volume 34, Issue 1, September 2003, Pages 3 – 24.

* Van der Waerden's theorem: For any given c and n, there is a number V, such that if V consecutive numbers are coloured with c different colours, then it must contain an arithmetic progression of length n whose elements are all the same colour.

Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory.

Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers

In 1938 he published a simpler derivation of Van Cittert's 1934 theorem on the coherence of radiation from distant sources, now known as the Van Cittert-Zernike theorem.

** Van der Waerden's theorem

Van Kampen's theorem for fundamental groups:

One can use Van Kampen's theorem to calculate fundamental groups for topological spaces that can be decomposed into simpler spaces.

Applying Van Kampen's theorem gives the result.

We now have enough information to apply Van Kampen's theorem.

In fact, modal logic is the fragment of first-order logic invariant under bisimulation ( Van Benthem's theorem ).

A theorem similar to van der Waerden's theorem is Schur's theorem: for any given c there is a number N such that if the numbers 1, 2, ..., N are coloured with c different colours, then there must be a pair of integers x, y

In 1903 Landau gave a much simpler proof than was then known of the prime number theorem and later presented the first systematic treatment of analytic number theory in the Handbuch der Lehre von der Verteilung der Primzahlen, or simply the Handbuch.

The theorem was first published by Thoralf Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme ( German: On the theory of associative number systems ) and later rediscovered by Emmy Noether.

* H. A. Lorentz, " The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light ," Amsterdammer Akademie der Wetenschappen 4 p. 176 ( 1896 ).

His work Theorie der algebraischen Functionen einer Veränderlichen ( with Dedekind ) established an algebraic foundation for Riemann surfaces, allowing a purely algebraic formulation of the Riemann-Roch theorem.