Degenerate Tur\'an problems for hereditary properties

Let $H$ be a graph and $t\geq s\geq 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $\lambda(G) = O\left(n^{1-1/s}\right)$ where $\lambda(G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.

In this paper, we consider a modification of the Turán-type question, where one forbids induced copies of a subgraph F. Without additional restrictions, this problem is trivial if F is not complete, because K n has ( n 2 ) edges and no induced F, whereas the problem is thoroughly investigated if F is a complete graph. More precisely, we study the maximum spectral radius that a graph may have if it has no induced K s,t and no copy (not necessarily induced) of a fixed forbidden subgraph H. We note that if χ(H) ≥ 3, then one may forbid H and have quadratically many edges, or one may forbid an induced copy of K s,t and have quadratically many edges. One of the main theorems in [8] shows that when one forbids both H and K s,t -induced at the same time, then a graph may not have quadratically many edges.
Theorem (Loh,Tait,Timmons [8]). Let s and t be integers and H be a graph. Then there is a constant C depending on s, t, and H such that if G is a graph on n vertices which has no copy of H as a subgraph and no copy of K s,t as an induced subgraph, then e(G) < Cn 2−1/s . Our main theorems are spectral strengthenings of the above theorem via the same inequality 2e(G)/n ≤ λ(G).
Finally, we note that this problem could be discussed in a more general context. A hereditary graph property is a family of graphs which is closed under isomorphisms and taking induced subgraphs. Given a hereditary property P, let P n denote the set of nvertex graphs in P. One may ask to find ex(n, P ) := max G∈P n e(G) and λ(n, P ) := max G∈P n λ(G). Nikiforov [11] found the asymptotics of these parameters similarly to the Erdős-Simonovits theorem for monotone graph properties. Let us note that both Erdos-Simonovits's and Nikiforov's theorems are informative only for problems with dense extremal graphs. Not surprisingly, extremal problems leading to sparse extremal graphs are harder and need special methods. As before, we call such problems degenerate.
In this note we focus on a degenerate extremal problem that we feel is quintessential for the area; our main result reads as follows. Theorem 1. Let t ≥ s ≥ 3 be integers, H be a graph, and K ≥ (R(H, K t )) 2/s R(H, K s ). If G is an H-free graph of order n and λ (G) then G contains an induced copy of K s,t .
Here, and throughout the rest of the paper, R(H, G) is the Ramsey number of H vs. G. One can question why in the premises of Theorem 1 the parameter s is at least 3, while 2 seems a more natural value. The reason is that for s = 2 we can prove a somewhat stronger estimate as stated in the theorem below. Theorem 2. Let r ≥ 2, t ≥ 2 be integers, and K ≥ R(K r , K t ). If G is a K r+1 -free graph of order n and λ (G) ≥ Kn 1/2 , then G contains an induced copy of K 2,t .
We note that Theorem 2 may be made more general by forbidding an arbitrary subgraph H instead of K r+1 . For ease of exposition, we will prove Theorem 2 only when H = K r+1 . It is clear from the proof of Theorem 1 how to generalize the result. The proofs of the two theorems differ at several points, so we shall keep them separate; they are proved in Section 2.
Finally, in Section 3 we will consider the specific case when H = C 5 and when an induced copy of K 2,t is forbidden. This will serve as an example of how, when more information about H is known, one can obtain close to tight estimates on the multiplicative constant. We will prove the following theorem.
Theorem 3. Let t ≥ 2 be an integer. If G is a C 5 -free graph with no induced copy of K 2,t , then For each integer t and prime power q for which t − 1 divides q 2 − 1, there is a bipartite K 2,t -free graph that is q-regular and has q 2 −1 t−1 vertices in each part (see Füredi [4]). Such a graph will have spectral radius 1 2 (t − 1)n + 1 where n is the number of vertices and so Theorem 3 is best possible up to a multiplicative factor of at most 2.
Before concluding our introduction, we mention the following corollary to Theorems 1 and 2 which implies one of the main results of [8] mentioned above.

Some notation
If G and H are graphs, we write H ≺ G to indicate that H is an induced subgraph of G.
Given a graph G, we write: for the set of independent r-sets of G and i r (G) for |I r (G) |; λ (G) for the largest eigenvalue of the adjacency matrix of G; -C 4 (G) for the number of 4-cycles of G; We also write: -( V r ) for the set of r-sets of a set V; -R(H, G) for the Ramsey number of H vs. G and R p,q for the Ramsey number of K p vs. K q .

Proofs of the main results
The following technical statement will be used in the proofs of both Theorems 1 and 2.

Proposition 5. If G is a graph of order n, then
Proof. Set for short V := V (G), and observe that Hence, On the other hand, writing CW 4 (G) for the number of closed walks of length 4, it is known that Finally, in view of the identity and Hofmeister's bound inequality (2) follows immediately from (4).

Proof of Theorem 2.
Suppose that t, r, K, and G satisfy the premises of the theorem. Note that for any pair X ∈ ( V 2 ), the graph G [Γ (X)] is K r -free; hence, Turán's theorem implies that and therefore, Summing this inequality over all pairs X ∈ ( V 2 ) and applying Proposition 5, we obtain Using Hofmeister's bound and some algebra, we find that That is to say, there is an , and so K 2,t ≺ G, completing the proof.
The proof of Theorem 1 is similar to the proof of Theorem 2, but needs a more technical approach; in particular, Turán's theorem does not apply as above. The focal point of the proof is the fact that if then G has an independent s-set I such that d (I) ≥ R(H, K t ); since G [Γ (I)] is H-free, it follows that K t ≺ G [Γ (I)], and hence K s,t ≺ G.
In turn, we deduce (5) along the following lines: we show that the premises of the theorem imply that G contains many copies of K 2,s , albeit not necessarily induced. However, the fact that G is H-free implies that for a positive proportion of these subgraphs of G their part of size s is an independent set. The requirement K ≥ (R(H, K t )) 2/s R(H, K s ) is sufficient to obtain (5) eventually.
To make this argument precise, we need three additional technical statements.

Proposition 6. If k ≥ 2 and G is a graph of order n with
Proof. The Power Mean inequality and inequality (2) imply that Hence, after simple algebra, we get Next, we find that To complete the proof, it is enough to note that if s > 2, the sum is precisely the number of K 2,s copies in G.

Proposition 8. Let s ≥ 2. If G is a H-free graph of order n, then
Proof. If n < R(H, K s ), then the inequality is trivial, so assume n ≥ R(H, K s ). Since G is H-free, any set of R(H, K s ) vertices must contain an independent set of size s. Each independent set of size s may be contained in at most n − s R(H, K s ) − s independent sets of size R(H, K s ). Therefore, Armed with the above propositions, we encounter no difficulty in proving Theorem 1. Suppose that s, t, H, K, and G satisfy the premises of the theorem. Note that n > s − 1, because n > λ (G) ≥ Kn 1−1/s and therefore n > K s > 2 s > s − 1. Further, for any X ∈ ( V 2 ), the graph G [Γ (X)] is H-free; hence, Proposition 8 implies that

Proof of Theorem 1.
Summing this inequality over all pairs X ∈ ( V 2 ) and double counting, we obtain On the other hand, Proposition 7 implies that G contains at least R(H, K t ) 2 R(H, K s ) s n s copies of K 2,s , that is to say, Combining this inequality with (7), we find that Therefore, inequality (5) holds; as shown above, it implies Theorem 1.

Forbidding C 5 and induced K 2,t
We begin this section with a general lemma that gives an upper bound on λ(G) that holds whenever G is H-free and has no induced K 2,t . Because we will be working with eigenvectors, it will be convenient to assume throughout this section that V(G) = {1, 2, . . . , n}. Furthermore, given a pair of vertices {i, j}, we will write d(i, j) rather than d({i, j}) and we do the same for Γ({i, j}). Lemma 9. Let t ≥ 2 be an integer and H be a graph with h ≥ 2 vertices. If G is an H-free graph of order n with no induced copy of K 2,t , then for any vertex x ∈ V(H), Additionally, if x and y is a pair of nonadjacent vertices in H, then R(H − x, K t ) can be replaced with R(H − x − y, K t ) in the estimate above.
Proof. Let x = (x 1 , . . . , x n ) be a non-negative eigenvector for the eigenvalue λ := λ(G) scaled to have 2-norm equal to 1. We have If 1 ≤ i < j ≤ n and {i, j} / ∈ E(G), then for any vertex x ∈ V(H), the common neighborhood Γ(i, j) cannot contain a copy of H − x or an independent set of size t, otherwise we find a copy of H or an induced copy of K 2,t . Therefore, Using this inequality, we have for any vertex x ∈ V(H), The double sum x i x j is at most n. This follows from two applications of the Cauchy-Schwarz inequality and the fact that x = 1. Therefore, counts pairs of vertices, say {z 1 , z 2 } ⊂ V(G), such that there is an edge {i, j} ∈ E(G) for which {z 1 , z 2 } ⊂ Γ(i, j). Suppose that this sum counts the same pair more than once. Let {z 1 , z 2 } ⊂ Γ(i, j) ∩ Γ(x, y). Without loss of generality, we may assume that i, j, and x are all distinct vertices. In this case, ijz 1 xz 2 i is a cycle of length 5 which is a contradiction. Thus, We conclude that λ(G) 2 ≤ 2tn + (n 2 + cn 3/2 ) 1/2 (3/8) 1/2 ≤ (2t + 0.375 1/2 )n + O(n 3/4 ).
Taking square roots completes the proof of Theorem 3.