Some asymptotic methods in combinatorics

Infinite Asymptotic Combinatorics

The following combinatorial theorems, some of which were known for every finite n in all infinite structures, are proved in ZFC for every infinite cardinal ν in all sufficiently large structures. (a) A new extension of Miller’s theorem [18]. (b) An upper bound of ρ on the list-conflict-free number of ρ-uniform families of sets which satisfy C(ρ, ν) for cardinals ν and ρ ≥ iω(n). (c) An upper bo...

Some Problems in Combinatorics

Let In = {1, 2, . . . , n} and x : In 7→ R be a map such that ∑ i∈In x(i) ≥ 0. (For any i, its image is denoted by x(i).) Let F = {J ⊂ In : |J | = k, and ∑ j∈J x(j) ≥ 0}. In [25] Manickam and Singhi have conjectured that |F| ≥ ( n−1 k−1 ) whenever n ≥ 4k and showed that the conclusion of the conjecture holds when k divides n. For any two integers r and ` let [r]` denote the smallest positive in...

Asymptotic bias of some election methods

Consider an election where N seats are distributed among parties with proportions p1, . . . , pm of the votes. We study, for the common divisor and quota methods, the asymptotic distribution, and in particular the mean, of the seat excess of a party, i.e. the difference between the number of seats given to the party and the (real) number Npi that yields exact proportionality. Our approach is to...

Some Asymptotic Methods for Strongly Nonlinear Equations

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 ...