Some extremal problems for hereditary properties of graphs

This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters.


Introduction
In this note we study problems stemming from the following one: What is the maximum number of edges a graph of order n, belonging to some hereditary property P.
Let us recall that a hereditary property is a family of graphs closed under taking induced subgraphs. For example, given a set of graphs F, the family of all graphs that do not contain any F ∈ F as an induced subgraph is a hereditary property, denoted as Her (F) .
It seems that the above classically shaped problem has been disregarded in the rich literature on hereditary properties, so we fill in this gap below.
Writing P n for the set of all graphs of order n in a property P, our problem now reads as: Given a hereditary property P, find ex (P, n) = max G∈Pn e (G) .
Finding ex (P, n) exactly seems hopeless for arbitrary P. A more feasible approach has been suggested by Katona, Nemetz and Simonovits in [7] who proved the following fact: Proposition 1 If P is a hereditary property, then the sequence ex (P, n) n 2 −1 ∞ n=1 is nonincreasing and so the limit always exists.
One of the aims of this paper is to establish π (P) for every P, but our main interest is in extremal problems about a different graph parameter, denoted by λ (α) (G) and defined as follows: for every graph G and every real number α ≥ 1, let Note first that λ (2) (G) is the well-studied spectral radius of G, and second, that λ (1) (G) is a another much studied parameter, known as the Lagrangian of G. So λ (α) (G) is a common generalization of two parameters that have been widely used in extremal graph theory.
The parameter λ (α) (G) has been recently introduced and studied for uniform hypergraphs first, by Keevash, Lenz and Mubayi in [6] and next by the author, in [13]. Here we shall study λ (α) (G) in the same setting as the number of edges in (1). Thus, given a hereditary property P , set As with ex (P, n) finding λ (α) (P, n) seems hopeless for arbitrary P. So, to begin with, the following theorem has been proved in [13] as an analog to Proposition 1.
Theorem 2 Let α ≥ 1. If P is a hereditary property, then the limit exists.
Thus, a natural question is to find λ (α) (P) for every P and every α ≥ 1. The main goal of this note to answer this question completely.

Main results
For notation and concepts undefined here, the reader is referred to [1].
Note first that every hereditary property P is trivially characterized by P = Her P , where P is the family of all graphs that are not in P; however, typically P can be given as P = Her (F) for some F that is only a small fraction of P.
Recall next that a complete r-partite graph is a graph whose vertices are split into r nonempty independent sets so that all edges between vertices of different classes are present. In particular, a 1-partite graph is just a set of independent vertices.
To characterize π (P) and λ (α) (P) we shall need two numeric parameters defined for every family of graphs F. First, let and second, let The parameters ω (F) and β (F) are quite informative about the hereditary property Her (F) , as seen first in the following observation.
Proof Suppose that ω (F) = 0. If ω (F) = 1, then P is empty, so we can suppose that ω (F) ≥ 2. This implies that β (F) > 0, as F contains K r for some r ≥ 2 and K r is a complete r-partite graph. If β (F) = 1, then F contains a graph G consisting of isolated vertices, say G is on s vertices.
Then either G contains a K r or an independent set on s vertices, both of which are forbidden. It turns out that β (F) ≥ 2, proving Proposition 3.
Clearly the study of (1) and (2) makes sense only if P is infinite and Proposition 3 provides necessary condition for this property of P. The following theorem completely characterizes π (P) .
To finish the proof we shall prove the opposite inequality. Let F ∈ F be a complete β-partite graph, known to exist by the definition of β (F) and let s be the maximum of the sizes of its vertex classes. Now assume that ε > 0 and set t = r (K r , K s ) , where r (K r , K s ) is the Ramsey number of K r vs. K s . If n is large enough and G ∈ P n satisfies then by the theorem of Erdős and Stone [5], G contains a subgraph G 0 = K β (t) , that is to say, a complete β-partite graph with t vertices in each vertex class. Since K r ∈ F, we see that G 0 contains no K r , hence each vertex class of G 0 contains an independent set of size s, and so G contains an induced subgraph K β (s) , which in turn contains an induced copy of F. Hence, if n is large enough and G ∈ P n , then This inequality implies that completing the proof.
We continue now with establishing λ (α) (P) for α > 1. The proof of our key Theorem 7 relies on several other results, some of which are stated within the proof itself. We give two other before the theorem. The first one follows from a result in [13], but for reader's sake we reproduce its short proof here.
We shall need also the following proposition (Proposition 29, [13]) whose proof we omit.
Proposition 6 Let α ≤ 1, k > 1 and G 1 and G 2 be graphs on the same vertex set. If G 1 and G 2 differ in at most k edges,then Here is the main theorem about λ (α) (P) .
Theorem 7 Let α > 1 and let F be a family of graphs. If the property P = Her (F) is infinite, then . Proof First note the inequality λ (α) (G) ≥ 2e (G) /n 2/α , which follows by taking (x 1 , . . . , x n ) = n −1/α , . . . , n −1/α in (2). So we see that and this, together with Theorem 4 gives λ (α) (P) = 1 if ω (F) = 0 and otherwise. To finish the proof we shall prove that For the purposes of this proof, write k r (G) for the number of r-cliques of G. Let F ∈ F be a complete β-partite graph, which exists by the definition of β (F) , and let s be the maximum of the sizes of its vertex classes.
We recall the following particular version of the Removal Lemma, one of the important consequences of the Szemerédi Regularity Lemma ( [15], [1]): Removal Lemma Let r ≥ 2 and ε > 0. There exists δ = δ (r, ε) > 0 such that if G is a graph of order n, with k r (G) < δn r , then there is a graph G 0 ⊂ G such that e (G 0 ) ≥ e (G) − εn 2 and k r (G 0 ) = 0.
In [11] we have proved the following theorem: Theorem A For all r ≥ 2, and ε > 0 there exists δ = δ (r, ε) > 0 such that if G a graph of order n with k r (G) > εn r , then G contains a K r (s) with s = ⌊δ log n⌋ . Now let ε > 0, choose δ = δ (β, ε) as in the Removal Lemma, and set t = r (K r , K s ) , where r (K r , K s ) is the Ramsey number of K r vs. K s . If G ∈ P n , then K β (t) G as otherwise we see as in proof of Theorem 4 that G contains an induced copy of F. So by Theorem A, if n is large enough, then k β (G) ≤ δn r . Now by the Removal Lemma there is a graph G 0 ⊂ G such that e (G 0 ) ≥ e (G) − εn 2 and k β (G 0 ) = 0. By Propositions 6 and 5, for n sufficiently large, we see that and hence, Since ε can be made arbitrarily small, we see that completing the proof of Theorem 7.
To complete the picture, we need to determine the dependence of λ (1) (P) on P. Using the the well-known idea of Motzkin and Straus, we come up with the following theorem, whose proof we omit Theorem 8 λ (1) (P)Let P be an infinite hereditary property. Then λ (1) (P) = 1 if P contains arbitrary large cliques, or λ (1) (P) = 1 − 1/r, where r is the size of the largest clique in P.

Concluding remarks
In a cycle of papers the author has shown that many classical exremal results like the Erdős-Stone-Bolloabs theorem [2], the Stability Theorem of Erdős [3,4] and Simonovits [14], and various saturation problems can be strengthened by recasting them for the largest eigenvalue instead of the number of edges; see [12] for overview and references.
The results in the present note and in [13] show that some of these results can be extended further for λ (α) (G) and α ≥ 1. A natural challenge here is to reprove systematically all of the above problems by substituting λ (α) (G) for the number of edges.