2 research outputs found
The Brown-Erd\H{o}s-S\'os Conjecture in finite abelian groups
The Brown-Erd\H{o}s-S\'{o}s conjecture, one of the central conjectures in
extremal combinatorics, states that for any integer if a 3-uniform
hypergraph on vertices contains no vertices spanning at least
edges, then the number of edges is We prove the conjecture for triple
systems coming from finite abelian groups
Maximum and minimum degree conditions for embedding trees
We propose the following conjecture: For every fixed , each graph of minimum degree at least and maximum
degree at least contains each tree with edges as a subgraph.
Our main result is an approximate version of the conjecture for bounded degree
trees and large dense host graphs. We also show that our conjecture is
asymptotically best possible. The proof of the approximate result relies on a
second result, which we believe to be interesting on its own. Namely, we can
embed any bounded degree tree into host graphs of minimum/maximum degree
asymptotically exceeding and , respectively, as long as
the host graph avoids a specific structure