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The (upper) clique number of a graph G, denoted omega(G), is the number of vertices in a maximum clique of G. Equivalently, it is the size of a largest clique or maximal ...

The middle levels conjecture, also known as revolving door conjecture, posits that the middle layer graph has a Hamilton cycle for every n>=1. The conjecture was proved by ...

Every closed three-manifold with finite fundamental group has a metric of constant positive scalar curvature, and hence is homeomorphic to a quotient S^3/Gamma, where Gamma ...

There exist infinitely many n>0 with p_n^2>p_(n-i)p_(n+i) for all i<n, where p_n is the nth prime. Also, there exist infinitely many n>0 such that 2p_n<p_(n-i)+p_(n+i) for ...

The first few numbers whose abundance absolute values are odd squares (excluding the trivial cases of powers of 2) are 98, 2116, 4232, 49928, 80656, 140450, 550564, 729632, ...

The probability that two elements P_1 and P_2 of a symmetric group generate the entire group tends to 3/4 as n->infty (Netto 1964, p. 90). The conjecture was proven by Dixon ...

If the Gauss map of a complete minimal surface omits a neighborhood of the sphere, then the surface is a plane. This was proven by Osserman (1959). Xavier (1981) subsequently ...

If P(z) is a power series which is regular for |z|<=1 except for m poles within this circle and except for z=+1, at which points the function is assumed continuous when only ...

A number n is called a k e-perfect number if sigma_e(n)=kn, where sigma_e(n) is the sum of the e-divisors of n.

A number n for which a shortest chain exists (which is also a Hansen chain) is called a Hansen number.