(mathematics) In additive combinatorics, a central result that indicates the approximate structure of sets whose sumset is small. It roughly states that if |A+A|/|A| is small, then A can be contained in a small generalized arithmetic progression.

(number theory) A theorem stating that the sequence of prime numbers contains arbitrarily long arithmetic progressions.

(mathematics) A fundamental combinatorial result of Ramsey theory, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure, and cannot be completely random.

(mathematics) In Ramsey theory, a theorem stating that, for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color.

(complex analysis) A theorem that gives a geometric relation between the roots of a polynomial P and the roots of its derivative P′. It states that the roots of P′ all lie within the convex hull of the roots of P.

(physics) A theorem that relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force F=-V'(x) on a massive particle moving in a scalar potential V(x).

A hypothesis that explains how flora and fauna migrated between Eurasia and Africa via a land bridge in the Levant region, based on the idea that there were periods of abundant rainfall lasting thousands of years in Africa and associated with a "wet-green Sahara" phase, during which larger lakes and more rivers existed.

(mathematics, complex analysis) The theorem that if a complex number is a root for a polynomial with real coefficients, the complex conjugate of that number will also be a root.

(mathematics) The two-dimensional counterpart of the ham sandwich theorem.