Asymptotic solution for a new class of forbiddenr-graphs

The inducibility of complete bipartite graphs

“…We will show that f is maximized at p = 1/2. (14) where x = (lp)/p. Let It follows that 1 is nondecreasing on [0,1/2], and since 1 is clearly symmetric about p = 1/2,1must have its maximum on [0,1] at 1/2.…”

“…Indeed, from (14) where h(x) = tx s + 1 -(t + s)X S + (t + s)x -t. It suffices to prove that h has just one root on (0, 1). This follows directly from the fact that h has the properties h(O) < 0, h(1) = 0, h'(1) < 0 and h"(x) < 0 for x E (0,1):…”

“…We begin by considering when for a given complete r-partite graph asymptotically optimal graphs can be found among complete r-partite graphs. The technique we apply (involving writing down a multivariate polynomial and considering its associated Lagrange multipliers) has been utilized in solutions to other combinatorial problems [6,11,13,14,15]. Proposition 7.…”

The Erdős‐Sós Conjecture for trees of diameter four

“…Prior to the formulation of the Conjecture, Erdős and Gallai [4] had shown that G embeds a path of length k À 1. (ii) Sidorenko [8] shows that G contains each tree of order k that has a vertex with at least k=2 À 1 leaf neighbors. Consequently, G contains every tree of diameter three.…”

Generalized Ramsey numbers for graphs

“…Even though this conjecture is still open, it has been shown that many trees are indeed ES-trees. For instance, stars (as an immediate consequence), paths (by Erdős and Gallai [50]), trees of order n that have a vertex with at least n/2 − 1 degree 1 neighbors (by Sidorenko [120]), trees of order n that have a vertex with at least n/2 − 2 degree 1 neighbors (by Eaton and Tiner [42]), double-stars (as an immediate corollary of Sidorenko's result), spiders of diameter at most 4 (by Woźniak [136]), trees of diameter at most 4 (by McLennan [99]), and spiders with three legs and spiders with no leg of length more than 4 (by Fan and Sun [53]). Two additional results were announced without being published: caterpillars are ES-trees (by Perles in 1990, as mentioned in [100]), and sufficiently large trees are ES-trees (by Ajtai et al [1]).…”