Most Probably Intersecting Families of Subsets

Let F be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality vertical bar F vertical bar <= 2(n-1). Suppose that vertical bar F vertical bar = 2(n-1) + i. Choose the members of F independently with probability p (delete them with probability 1 - p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing F appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all [n/2]-element subsets. We determine the most probably inclusion-free family too, when the number of members is (n([n/2])) + 1

Factorization of Laurent series over commutative rings

We generalize the Wiener-Hopf factorization of Laurent series to more general commutative coefficient rings, and we give explicit formulas for the decomposition. We emphasize the algebraic nature of this factorization.Comment: 8 page

Constructing Union-Free pairs of K-Element subsets

It is proved that one can choose [1/2(n/k)] disjoint pairs of k-element subsets of an n-element set in such a way that the unions of the pairs are all different, supposing that n > n(k)

Calculating the Hubble diagram by perturbation theory

The effect of density fluctuations upon light propagation is calculated perturbatively in a matter dominated irrotational universe. The starting point is the perturbed metric (second order in the perturbation strength), while the output is the Hubble diagram. Density fluctuations cause this diagram to broaden to a strip. Moreover, the shift of the diagram mimics accelerated expansion