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As shown by Schur (1916), the Schur number S(n) satisfies S(n)<=R(n)-2 for n=1, 2, ..., where R(n) is a Ramsey number.

A graph G is the edge graph of a polyhedron iff G is a simple planar graph which is 3-connected.

Let G=(V,E) be a finite graph, let Omega be the set Omega={0,1}^E whose members are vectors omega=(omega(e):e in E), and let F be the sigma-algebra of all subsets of Omega. A ...

Connecting the centers of touching spheres in a three-dimensional Apollonian gasket by edges given a graph known as the Apollonian network. This process is illustrated above ...

The folded n-cube graph, perhaps better termed "folded hypercube graph," is a graph obtained by merging vertices of the n-hypercube graph Q_n that are antipodal, i.e., lie at ...

The axiom of Zermelo-Fraenkel set theory which asserts that sets formed by the same elements are equal, forall x(x in a=x in b)=>a=b. Note that some texts (e.g., Devlin ...

The axiom of Zermelo-Fraenkel set theory which asserts the existence for any set a and a formula A(y) of a set x consisting of all elements of a satisfying A(y), exists x ...

Let A^~, B^~, ... be operators. Then the commutator of A^~ and B^~ is defined as [A^~,B^~]=A^~B^~-B^~A^~. (1) Let a, b, ... be constants, then identities include [f(x),x] = 0 ...

A technique in set theory invented by P. Cohen (1963, 1964, 1966) and used to prove that the axiom of choice and continuum hypothesis are independent of one another in ...

Let (A,<=) and (B,<=) be disjoint totally ordered sets with order types alpha and beta. Then the ordinal sum is defined at set (C=A union B,<=) where, if c_1 and c_2 are both ...