Improved bounds for the extremal number of subdivisions

Let $H_t$ be the subdivision of $K_t$. Very recently, Conlon and Lee have proved that for any integer $t\geq 3$, there exists a constant $C$ such that $\text{ex}(n,H_t)\leq Cn^{3/2-1/6^t}$. In this paper, we prove that there exists a constant $C'$ such that $\text{ex}(n,H_t)\leq C'n^{3/2-\frac{1}{4t-6}}$.


Introduction
For a graph H, the extremal function ex(n, H) is defined to be the maximal number of edges in an H-free graph on n vertices. This function is well understood for graphs H with chromatic number at least three by the Erdős-Stone-Simonovits theorem. [3,5] However, for bipartite graphs H, much less is known. For a survey on the subject, see [7]. One of the few general results, proved by Füredi [6], and reproved by Alon, Krivelevich and Sudakov [1] is the following.
Theorem 1 (Füredi, Alon-Krivelevich-Sudakov). Let H be a bipartite graph such that in one of the parts all the degrees are at most r. Then there exists a constant C such that ex(n, H) ≤ Cn 2−1/r . [2] have conjectured that the only case when this is tight up to the implied constant is when H contains a K r,r (it is conjectured [8] that ex(n, K r,r ) = Ω(n 2−1/r )), and that for other graphs H there exists some δ > 0 such that ex(n, H) = O(n 2−1/r−δ ).

Conlon and Lee
The subdivision of a graph L is the bipartite graph with parts V (L) and E(L) (the vertex set and the edge set of the graph L, respectively) where v ∈ V (L) is joined to e ∈ E(L) if v is an endpoint of e. It is easy to see that any C 4 -free bipartite graph in which every vertex in one part has degree at most two is a subgraph of H t for some positive integer t, where H t is the subdivision of K t . Conlon and Lee have verified their conjecture in the r = 2 case by proving the following result.
They have observed the lower bound ex(n, H t ) ≥ c t n 3/2− t−3/2 t 2 −t−1 coming from the probabilistic deletion method, and have asked for an upper bound of the form ex(n, H) ≤ C t n 3/2−δt , where 1/δ t is bounded by a polynomial in t. We can prove such a bound even for a linear δ t . Theorem 3. For any integer t ≥ 3, there exists a constant C t such that ex(n, It would be very interesting to know whether or not this bound is tight up to the implied constant. It certainly is tight for t = 3 as ex(n, C 6 ) = Θ(n 4/3 ).
We can in fact prove a slightly stronger result. For integers s ≥ 1 and t ≥ 3, let L s,t be the graph which is a K s+t−1 with the edges of a K s removed. That is, the vertex set of L s,t is S ∪ T where S ∩ T = ∅, |S| = s and |T | = t − 1, and xy is an edge if and only if x ∈ T or y ∈ T . Let L ′ s,t be the subdivision of L s,t .
Theorem 4. For any two integers s ≥ 1 and t ≥ 3, there exists a constant C s,t such that ex(n, L ′ s,t ) ≤ C s,t n 3/2− 1 4t−6 .
This result certainly implies Theorem 3 as L 1,t = K t . Moreover, we can apply Theorem 4 to obtain good bounds on the extremal number of the subdivision of the complete bipartite graph K a,b as well. Let us write H a,b for the subdivision of K a,b . Conlon and Lee [2, Theorem 4.2] have proved that for any 2 ≤ a ≤ b there exists a constant C such that ex(n, H a,b ) ≤ Cn 3/2− 1 12b . They have also observed the lower bound ex(n, H a,b ) = Ω a,b (n 3/2− a+b−3/2 2ab−1 ) (which follows from the probabilistic deletion method). Hence their upper bound is reasonably close to best possible when a = b, but is weak when b is much larger then a.
Since K a,b is a subgraph of L b,a+1 , Theorem 4 implies the following result, by taking s = b and t = a + 1.

Proof of Theorem 4
We shall use the following lemma of Conlon and Lee [2, Lemma 2.3], which is a slight modification of a result of Erdős and Simonovits [4]. Let us say that a graph G . Moreover, following Conlon and Lee, we say that a bipartite graph G with a bipartition A ∪ B is balanced if 1 2 |B| ≤ |A| ≤ 2|B|. Lemma 6. For any positive constant α < 1, there exists n 0 such that if n ≥ n 0 , C ≥ 1 and G is an n-vertex graph with at least Cn 1+α edges, then G has a K- 10 m 1+α and K = 60 · 2 1+1/α 2 . This reduces Theorem 4 to the following.
Theorem 7. For every K ≥ 1, and positive integers s ≥ 1, t ≥ 2, there exists a constant c = c(s, t, K) with the following property. Let n be sufficiently large and let G be a balanced bipartite graph with bipartition A ∪ B, |B| = n such that the degree of every vertex of G is between δ and Kδ, for some δ ≥ cn t−2 2t−3 . Then G contains a copy of L ′ s,t .
Given a bipartite graph G with bipartition A ∪ B, the neighbourhood graph is the weighted graph W G on vertex set A where the weight of the pair uv is d G (u, v) = |N G (u) ∩ N G (v)|. Here and below N G (v) denotes the neighbourhood of the vertex v in the graph G. For a subset U ⊂ A, we write W (U) for the total weight in U, ie.
. We shall use the following simple lemma of Conlon and Lee [2, Lemma 2.4].
Lemma 8. Let G be a bipartite graph with bipartition A ∪ B, |B| = n, and minimum degree at least δ on the vertices in A. Then for any subset U ⊂ A with δ|U| ≥ 2n, In other words, the conclusion of Lemma 8 is that In the next definition, and in the rest of this paper, for a weighted graph W on vertex set A, if u, v ∈ A, then W (u, v) stands for the weight of uv. Moreover, we shall tacitly assume throughout the paper that s ≥ 1 and t ≥ 3 are fixed integers.
Definition 9. Let W be a weighted graph on vertex set A and let u, v ∈ A be distinct. We say that uv is a light edge if 1 ≤ W (u, v) < s+t−1 2 and that it is a heavy edge if Note that if there is a K s+t−1 in W G formed by heavy edges, then clearly there is an L s,t in W G formed by heavy edges, therefore there is an L ′ s,t in G. The next lemma is one of our key observations. Lemma 10. Let G be an L ′ s,t -free bipartite graph with bipartition A ∪ B, |B| = n and suppose that W (A) ≥ 8(s + t) 2 n. Then the number of light edges in W G is at least formed by heavy edges. Thus, by Turán's theorem, the number of light edges in N G (b i ) is at least Since every light edge is present in at most s+t−1 2 of the sets N G (b i ), it follows that the total number of light edges is at least Corollary 11. Let G be an L ′ s,t -free bipartite graph with bipartition A ∪ B, |B| = n, and minimum degree at least δ on the vertices in A. Then for any subset U ⊂ A with |U| ≥ 8(s+t)n δ and |U| ≥ 2, the number of light edges in W G [U] is at least We are now in a position to complete the proof of Theorem 7.
Proof of Theorem 7. Let c be specified later and suppose that n is sufficiently large. Assume, for contradiction, that G is L ′ s,t -free. We shall find distinct u 1 , . . . , u t−1 ∈ A with the following properties.
As n is sufficiently large, we have |A| ≥ n/2 ≥ 8(s+t)n δ , therefore by Corollary 11 there are at least δ 2 8(s+t) 3 n |A| 2 light edges in A, so we may choose u 1 ∈ A such that the number of light edges u 1 v is at least δ 2 8(s+t) 3 n (|A| − 1) ≥ δ 2 32(s+t) 3 n |A|. Now suppose that 2 ≤ i ≤ t − 1, and that u 1 , . . . , u i−1 have been constructed satisfying (i),(ii) and (iii). Let U 0 be the set of vertices v ∈ A with the property that u j v is a light edge for every j ≤ i − 1. By (iii), we have |U 0 | ≥ ( δ 2 32(s+t) 3 n ) i−1 |A|.
Kδ. But note that for sufficiently large n, we have Kδ because δ = o((δ 2 /n) t−2 n) and δ = o((δ 2 /n)n). Thus, But for sufficiently large c = c(s, t, K), we have 1 2 ( δ 2 32(s+t) 3 n ) i−1 |A| ≥ 8(s+t)n δ . Indeed, this is obvious when δ 2 ≥ 32(s + t) 3 n, and otherwise, using δ ≥ cn t−2 2t−3 , we have 1 2 Thus, by Corollary 11, there exists some u i ∈ U with at least δ 2 8(s+t) 3 n (|U| − 1) ≥ ( δ 2 32(s+t) 3 n ) i |A| light edges adjacent to it in U. This completes the recursive construction of the vertices {u j } 1≤j≤t−1 . By (iii) for i = t − 1, there is a set V ⊂ A consisting of at least ( δ 2 32(s+t) 3 n ) t−1 |A| vertices v such that for every j ≤ t − 1, u j v is a light edge. We shall now prove that there exist distinct v 1 , . . . , v s ∈ V such that N G (u i ) ∩ N G (u j ) ∩ N G (v k ) = ∅ for all i = j, and N G (u i ) ∩ N G (v j ) ∩ N G (v k ) = ∅ for all j = k. It is easy to see that this suffices since then there is a copy of L ′ s,t in G, which is a subdivision of the copy of L s,t in W G whose vertices are v 1 , . . . , v s , u 1 , . . . , u t−1 .
We shall now choose v 1 , . . . , v s one by one. Since every u i u j is a light edge, the number of those v ∈ A with N G (u i ) ∩ N G (u j ) ∩ N G (v) = ∅ for some i = j is at most Kδ. Moreover, given any choices for v 1 , . . . , v k−1 ∈ V , as each u i v j is a light edge, the number of those v ∈ A with N G (u i )∩N G (v j )∩N G (v) = ∅ for some i, j is at most (t−1)(k −1) s+t−1 2 Kδ. Therefore as long as |V | > t−1 2 s+t−1 2 Kδ, suitable choices for v 1 , . . . , v s can be made. Since |V | ≥ ( δ 2 32(s+t) 3 n ) t−1 |A|, this last inequality holds for large enough c = c(s, t, K).