The Erdős–Fuchs theorem is a result in number theory concerning the distribution of prime numbers. It provides a characterization of the divisibility of numbers in the context of prime factors.

Dilworth's theorem is a result in order theory, a branch of mathematics that studies the properties of ordered sets. The theorem states that in any finite partially ordered set (poset), the size of the largest antichain (a subset of elements in which no two elements are comparable) is equal to the smallest number of chains (totally ordered subsets) that can cover the poset. In more formal terms: - Let \( P \) be a finite poset.

Baranyai's theorem is a result in combinatorial design theory, specifically within the area of finite set systems. It is named after its discoverer, Zsolt Baranyai. The theorem deals with the partitioning of a complete graph into smaller structures, namely, it provides conditions under which it is possible to partition the complete graph on a certain number of vertices into disjoint complete subgraphs of smaller sizes.

The Whitney extension theorem is a fundamental result in the field of analysis and differential geometry, concerning the extension of functions defined on a subset of a Euclidean space to the entire space while preserving certain properties.

The Unique Homomorphic Extension Theorem is a result in the field of algebra, particularly concerning rings and homomorphisms. It typically states that if you have a ring \( R \) and a subring \( S \), along with a homomorphism defined on \( S \), then there exists a unique (in the case of certain conditions) homomorphic extension of this mapping up to the whole ring \( R \).

Trudinger's theorem, often discussed in the context of variational calculus and partial differential equations, refers to a result concerning minimization problems for integral functionals that involve "non-standard" growth conditions. Specifically, it addresses the existence of solutions to certain minimization problems that contain terms with exponential growth.

The Sturm-Picone comparison theorem is a fundamental result in the theory of ordinary differential equations, especially in the context of second-order linear differential equations.

The Sturm separation theorem is a fundamental result in real analysis and the theory of differential equations, particularly in the context of Sturm-Liouville problems. It deals with the properties of the roots of Sturm polynomials, which are solutions to a certain class of linear differential equations.

The Stone–Weierstrass theorem is a fundamental result in analysis that provides conditions under which a set of functions can approximate continuous functions on a compact space. It generalizes the Weierstrass approximation theorem, which specifically addresses polynomial functions. Here is a more formal statement of the theorem: Let \( X \) be a compact Hausdorff space, and let \( C(X) \) denote the space of continuous real-valued functions on \( X \).

Stahl's theorem is a result in the field of mathematics, specifically in complex analysis and the theory of analytic functions. It deals with the boundary behavior of meromorphic functions and their poles.

The Silverman-Toeplitz theorem is a result in functional analysis and operator theory concerning the convergence of certain types of series of bounded linear operators. Specifically, it addresses the behavior of a series of projections in a Hilbert space. The theorem can be stated as follows: Let \( H \) be a Hilbert space and let \( \{ P_n \} \) be a sequence of orthogonal projections in \( H \).

The Shift Theorem, often associated with the field of signal processing and control theory, provides a useful relationship between the time domain and the frequency domain of a signal. It primarily refers to how a time shift in a signal affects its Fourier transform.

Sard's theorem is a result in differential topology that pertains to the behavior of smooth functions between manifolds. Specifically, it addresses the notion of the image of a smooth function and the measure of its critical values.

The Remez inequality is a result in approximation theory that provides a bound on the deviation of a continuous function from its best approximation by a polynomial. Specifically, it relates the norm of a polynomial approximation to the maximum deviation of the approximated function over a given interval.

The Rellich–Kondrachov theorem is a significant result in functional analysis and the theory of differential equations, particularly in the context of Sobolev spaces. It essentially states conditions under which the embedding of Sobolev spaces into Lp spaces is compact.

The Rademacher–Menchov theorem is a result in the field of measure theory and functional analysis. It is particularly significant in the study of series of functions, specifically in the context of rearrangement of series in Banach spaces.

The Poincaré inequality is a fundamental result in mathematical analysis and partial differential equations. It provides a bound on the integral of a function in terms of the integral of its derivative.

The Picard–Lindelöf theorem, also known as the Picard existence theorem or the Picard-Lindelöf theorem, is a fundamental result in the theory of ordinary differential equations (ODEs). It provides conditions under which a first-order ordinary differential equation has a unique solution in a specified interval.

The Peano existence theorem, often referred to in the context of ordinary differential equations (ODEs), is a fundamental result that provides conditions under which solutions to certain initial value problems exist.