56 research outputs found
Strong Jumps and Lagrangians of Non-Uniform Hypergraphs
The hypergraph jump problem and the study of Lagrangians of uniform
hypergraphs are two classical areas of study in the extremal graph theory. In
this paper, we refine the concept of jumps to strong jumps and consider the
analogous problems over non-uniform hypergraphs. Strong jumps have rich
topological and algebraic structures. The non-strong-jump values are precisely
the densities of the hereditary properties, which include the Tur\'an densities
of families of hypergraphs as special cases. Our method uses a generalized
Lagrangian for non-uniform hypergraphs. We also classify all strong jump values
for -hypergraphs.Comment: 19 page
-free families in the Boolean lattice
For a family of subsets of [n]=\{1, 2, ..., n} ordered by
inclusion, and a partially ordered set P, we say that is P-free
if it does not contain a subposet isomorphic to P. Let be the
largest size of a P-free family of subsets of [n]. Let be the poset with
distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean
lattice. We show that where . We also prove that the largest -free
family of subsets of [n] having at most three different sizes has at most
2.20711N members.Comment: 18 pages, 2 figure
- …