A weighted Turán sieve method

Turán problems for integer-weighted graphs

A multigraph is (k, r)-dense if every k-set spans at most r edges. What is the maximum number of edges exN(n, k, r) in a (k, r)-dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r > k) by determining a constant m = m(k, r) and showing that exN(n, k, r) = m ( n 2 ) +O(n), thus giving a generalization of Turán’s theorem. ...

A Method for Defuzzication by Weighted Distance

A Sieve Method for Factoring Numbers

Sieve Method and Landau Problems

At the 1912 Fifth International Congress of Mathematicians (Cambridge), Landau mentioned four unattackable problems: Goldbach Conjecture; Twin Prime Conjecture; for each n, there is a prime p such that n2 < p < (n+1)2; there are infinitely many primes p of the form p = n2+1. As Erdős [6, preface] maintained five years ago, we still do not have a satisfactory solution for each of them. For large...

A hypergraph regularity method for generalized Turán problems

Wedescribe amethod that we believemay be foundational for a comprehensive theory of generalized Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed.We demonstrate the power of the me...