Dade’s ordinary conjecture implies the Alperin–McKay conjecture

The Füredi–Hajnal Conjecture Implies the Stanley–Wilf Conjecture

We show that the Stanley–Wilf enumerative conjecture on permutations follows easily from the Füredi–Hajnal extremal conjecture on 0-1 matrices. We apply the method, discovered by Alon and Friedgut, that derives an (almost) exponential bound on the number of some objects from a (almost) linear bound on their sizes. They proved by it a weaker form of the Stanley–Wilf conjecture. Using bipartite g...

The Graham Conjecture Implies the Erdös-turán Conjecture

Erdös and Turán once conjectured that any set A ⊂ N with

The Bochner-riesz Conjecture Implies the Restriction Conjecture

Vojta’s Conjecture Implies the Batyrev-manin Conjecture for K3 Surfaces

Vojta’s Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic varieties with nonnegative Kodaira dimension. We use this to deduce, from Vojta’s Conjectures, conjectures of Batyrev-Manin and Manin on the distribution of rational...

The Abc Conjecture Implies Roth’s Theorem and Mordell’s Conjecture

We present in a unified way proofs of Roth’s theorem and an effective version of Mordell’s conjecture, using the ABC conjecture. We also show how certain stronger forms of the ABC conjecture give information about the type of approximation to an algebraic number.