Search

In this article we approach some of the basic questions in topological dynamics, concerning periodic points, transitivity, the shadowing and the gluing orbit properties, in the context of C 0 incompressible flows generated by Lipschitz vector fields. We prove that a C 0 -generic incompressible and fixed-point free flow satisfies the periodic shadowing property, it is transitive and has a dense set of periodic points in the non-wandering set. In particular, a C 0 -generic fixed-point free incompressible flow satisfies the reparametrized gluing orbit property. We also prove that C 0 -generic incompressible flows satisfy the general density theorem and the weak shadowing property, moreover these are transitive. [ABSTRACT FROM AUTHOR]

Let n > k > t ≥ j ≥ 1 be integers. Let X be an n -element set, ( X k ) the collection of its k -subsets. A family F ⊂ ( X k ) is called t -intersecting if | F ∩ F ′ | ≥ t for all F , F ′ ∈ F. The j 'th shadow ∂ j F is the collection of all (k − j) -subsets that are contained in some member of F. Estimating | ∂ j F | as a function of | F | is a widely used tool in extremal set theory. A classical result of the second author (Theorem 1.3) provides such a bound for t -intersecting families. It is best possible for | F | = ( 2 k − t k ). Our main result is Theorem 1.4 which gives an asymptotically optimal bound on | ∂ j F | / | F | for | F | slightly larger, e.g., | F | > 3 2 ( 2 k − t k ). We provide further improvements for | F | very large as well. [ABSTRACT FROM AUTHOR]

A fundamental result in extremal set theory is Katona's shadow intersection theorem, which extends the Kruskal–Katona theorem by giving a lower bound on the size of the shadow of an intersecting family of k -sets in terms of its size. We improve this classical result and a related result of Ahlswede, Aydinian and Khachatrian by proving tight bounds for families that can be quite small. For example, when k = 3 our result is sharp for all families with n points and at least 3 n − 7 triples. Katona's theorem was extended by Frankl to families with matching number s. We improve Frankl's result by giving tight bounds for large n. [ABSTRACT FROM AUTHOR]