The Erdös–Sós conjecture for spiders

A classical result on extremal graph theory is the Erdös–Gallai theorem: if a graph on n   vertices has more than (k-1)n2 edges, then it contains a path of k edges. Motivated by the result, Erdös and Sós conjectured that under the same condition, the graph should contain every tree of k edges. A spider is a rooted tree in which each vertex has degree one or two, except for the root. A leg of a spider is a path from the root to a vertex of degree one. Thus, a path is a spider of 1 or 2 legs. From the motivation, it is natural to consider spiders of 3 legs. In this paper, we prove that if a graph on n   vertices has more than (k-1)n2 edges, then it contains every k-edge spider of 3 legs, and also, every k-edge spider with no leg of length more than 4, which strengthens a result of Woźniak on spiders of diameter at most 4.