Sieve methods in combinatorics

Sieve methods in combinatorics

We develop the Turán sieve and a ‘simple sieve’ in the context of bipartite graphs and apply them to various problems in combinatorics. More precisely, we provide applications in the cases of characters of abelian groups, vertex-colourings of graphs, Latin squares, connected graphs, and generators of groups. In addition, we give a spectral interpretation of the Turán sieve. © 2004 Elsevier Inc....

Sieve methods

Sieve Methods

Preface Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of ...

Algebraic Methods in Combinatorics

1. (A result of Bourbaki on finite geometries, from Răzvan) Let X be a finite set, and let F be a family of distinct proper subsets of X. Suppose that for every pair of distinct elements in X, there is a unique member of F which contains both elements. Prove that |F| ≥ |X|. Solution: Let X = [n] and F = {A1, . . . , Am}. We need to show that n ≤ m. Define the m × n incidence matrix A over R by ...

Probabilistic Methods in Combinatorics

Remark. This is also known as the Lubell-Yamamoto-Meshalkin inequality. Solution: Generate a random permutation σ = (x1, . . . , xM ) of [M ]. Let each Ei be the event that the set Ai appears as the initial segment of σ, i.e., that {x1, . . . , xai} = Ai. Since none of the Ai contain each other, the Ei are mutually exclusive. But each P [Ei] = ( M ai )−1 , so the LHS is P [∪Ei], and probabiliti...