On the weight of Berge-$F$-free hypergraphs

For a graph $F$, we say a hypergraph is a Berge-$F$ if it can be obtained from $F$ by replacing each edge of $F$ with a hyperedge containing it. A hypergraph is Berge-$F$-free if it does not contain a subhypergraph that is a Berge-$F$. The weight of a non-uniform hypergraph $\mathcal{H}$ is the quantity $\sum_{h \in E(\mathcal{H})} |h|$. Suppose $\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this short note, we prove that as long as every edge of $\mathcal{H}$ has size at least the Ramsey number of $F$ and at most $o(n)$, the weight of $\mathcal{H}$ is $o(n^2)$. This result is best possible in some sense. Along the way, we study other weight functions, and strengthen results of Gerbner and Palmer; and Gr\'osz, Methuku and Tompkins.


Introduction
Generalizing the notion of hypergraph cycles due to Berge, the authors Gerbner and Palmer [8] introduced the so-called Berge hypergraphs. Given a graph F , we say that a hypergraph H is Berge-F if there is a bijection f : E(F ) → E(H) such that for every e ∈ E(F ) we have e ⊆ f (e). Equivalently, H is Berge-F if we can embed a distinct graph edge into each hyperedge of H to obtain a copy of F . Note that for a fixed F there are many different hypergraphs that are Berge-F , and a fixed hypergraph H can be Berge-F for many different graphs F . Let B c (F ) denote the set of all c-uniform hypergraphs that are Berge-F .
Most of these results deal with the uniform case, but some also examine non-uniform hypergraphs. Note that replacing a hyperedge with a larger hyperedge containing it never removes a copy of Berge-F , but may add a copy. Thus, to build a Berge-F -free hypergraph that maximizes the number of hyperedges, one picks small hyperedges. To make large hyperedges more attractive, one can assign a weight to each hyperedge that increases with the size of the hyperedge.
Győri [12] proved that if H is a Berge-triangle-free hypergraph, then h∈E(H) (|h|−2) ≤ n 2 /8 if n is large enough. Note that this result is about a multi-hypergraph H, thus h∈E(H) |h| can be arbitrarily large by taking a hyperedge of size 2 an arbitrary number of times. In [15], the authors showed that for a Berge-C 4 -free multi-hypergraph H we have h∈E(H) (|h| − 3) ≤ 12 √ 2n 3/2 + O(n) and they gave a construction of a Berge-C 4 -free multi-hypergraph with approximately n 3/2 /8 hyperedges. The upper bound was improved by Gerbner and Palmer [8] to √ 6n 3/2 /2, while the lower bound was improved to (1 + o(1)n 3/2 /(3 √ 3). For arbitrary cycles, Győri and Lemons [16] proved that if H is either a Berge-C 2k -free or Berge-C 2k+1 -free hypergraph on n vertices and every hyperedge in H has size at least 4k 2 , then h∈E(H) |h| = O(n 1+1/k ).
Gerbner and Palmer [8] proved the following general result about Berge-F -free hypergraphs.
Theorem 1 (Gerbner and Palmer [8]). Let F be a graph and let H be a Berge-Ffree hypergraph on n vertices. If every hyperedge in H has size at least |V (F )|, then h∈E(H) |h| = O(n 2 ).
We strengthen Theorem 1 in Theorem 3 by showing that the statement still holds if one replaces |h| with |h| 2 in the above sum; moreover, our proof is much simpler compared to the proof of Gerbner and Palmer in [8]. For uniform hypergraphs, the above theorem states that for any graph F and Berge-F -free r-uniform hypergraph H we have |E(H)| = O(n 2 ) provided r is large enough. Grósz, Methuku and Tompkins showed that, in fact, |E(H)| = o(n 2 ) for large enough r. This is stated more precisely in the following theorem. Given two r-uniform hypergraphs F and K, let R (r) (F, K) denote the 2-color runiform Ramsey number of F and K. Let R(F, K) := R (2) (F, K), R (r) (F ) := R (r) (F, F ) and R(F ) := R(F, F ).
We improve this theorem in Theorem 4. Let us return to non-uniform hypergraphs. So far, we have only added up the sizes of the hyperedges. Here we will change the weight function and consider h∈E(H) |h| c for different values of c.
As an immediate corollary to Theorem 3, we can see that for any c ≥ 2 and any Berge-F -free hypergraph that satisfies the conditions of Theorem 3, we have Furthermore, this result is trivially sharp as can be seen by considering any hypergraph with at least one edge of size Ω(n). Interestingly, the next theorem suggests that large edges are necessary for such a weighted sum to reach this upper bound.
Theorem 4. Let F be a fixed graph, 2 ≤ c ≤ |V (F )| be an integer and let K ∈ B c (F ). Let H be a Berge-F -free hypergraph on n vertices such that every edge of H has size at least R (c) (K) and at most o(n Combining Theorem 3 and Theorem 4, we can show the sum of the sizes of the edges of a Berge-F -free hypergraph is o(n 2 ) provided that all the hyperedges are large enough (but not necessarily growing with n), presenting another improvement of Theorem 1 and Theorem 2. In Theorem 5, the smallest possible size of edges allowed in H must grow with the forbidden graph F : Indeed, let r be an integer and assume r | n. Let a vertex set on n vertices be partitioned into n/r singletons and n/r sets of size r − 1. Let H be the r-uniform hypergraph consisting of all the edges that contain one singleton and one (r − 1)-set. Then it is easily verified that H is an r-uniform Berge-K r -free hypergraph, but h∈E(H) |h|= n 2 /r. In fact, it was shown by Grósz, Methuku and Tompkins [11] that there are (ω(F ) − 1) 2 -uniform Berge-F -free hypergraphs of size Ω(n 2 ), where ω(F ) denotes the clique number of F . It is an interesting open problem to determine the smallest uniformity when Ω(n 2 ) drops to o(n 2 ).
On the other hand, it is worth noting that the bound o(n 2 ) is close to being best possible: Erdős, Frankl and Rödl [4] constructed r-uniform hypergraphs with more than n 2−ε hyperedges for any ε, and with the property that there are no 3 hyperedges on 3(r − 1) vertices. Observe that a Berge-triangle is on at most 3(r − 1) vertices, hence those hypergraphs are also Berge-triangle-free.

Notation
In the rest of the paper, we use the following notation. We will refer to a c-uniform hypergraph as a c-graph and its hyperedges as c-edges. For a set S of vertices, let Γ (c) (S) denote the c-graph with vertex set S, and whose edge-set is the set of all c-tuples contained in S. all the c-tuples contained in at least one hyperedge of H. For ease of notation, we will let Γ(H) := Γ (2) (H). We will denote by K (c) n the complete c-uniform hypergraph on n vertices.
We will need to consider both general Berge copies of F and also uniform Berge copies of F . Recall that B c (F ) denotes the family of c-uniform Berge-F hypergraphs, and let " B c (F ) denote the family of c-uniform Berge-F multi-hypergraphs. Note by not allowing Berge-F hypergraphs to contain unnecessary isolated vertices, we have that B c (F ) and " B c (F ) are finite.

Proof of Theorem 4
First, we need to establish a lemma that will allow us to find certain uniform Berge structures in non-uniform Berge structures. There is a straightforward generalization of Berge copies of graphs: Berge copies of hypergraphs. More precisely, if K and H are both (multi-)hypergraphs, we say H is a Berge-K if there is a bijection f : E(K) → E(H) such that for every e ∈ E(K), we have e ⊆ f (e). This version has already been studied, see [1,3,10]. Here we follow an argument due to Grósz, Methuku and Tompkins [11]. We wish to apply the hypergraph removal lemma to the c-shadow of H for some hypergraph K ∈ B c (F ). To this end, we prove the following claim. Fix K ∈ B c (F ). By Claim 10 and the hypergraph removal lemma, there exists a set R of c-edges such that each copy of K in Γ (c) (H) contains at least one c-edge in R and |R|= o(n c ). We will call a c-edge in Γ (c) (H) special if it contained in R and is contained in at most |E(F )| − 1 hyperedges of H. Note that the special c-edges here play a similar but slightly different role than the blue edges in the proof of Theorem 3. Let R ′ be the set of all the special c-edges. Of course, R ′ ⊆ R.
Claim 11. Let h ∈ E(H) be an arbitrary hyperedge. Then any subset S ⊆ h of size R (c) (K) contains a special c-edge.
Proof. Assume for a contradiction that there is a set S ⊆ h of size R (c) (K) which contains no special c-edge. In other words, every c-edge of R contained in S is in at least |E(F )| hyperedges. By the definition of R, Γ (c) (S) \ R cannot contain a copy of K. Since Γ (c) (S) is a complete hypergraph, we may apply Ramsey's theorem with the c-edges in E(Γ (c) (S)) \ R colored red and those in E(Γ (c) (S)) ∩ R colored blue. There cannot be a copy of K in the red c-edges, so there must be a copy using blue c-edges, but since each blue c-edge is in |E(F )| hyperedges of H, there is a Berge-K in H, and so by Lemma 8, there is a Berge-F in H, a contradiction. Now we provide a lower bound on the number of special c-edges contained in a hyperedge of H.

Claim 12.
There exists a constant ℓ < 1 such that for any hyperedge h ∈ E(H), we have that Proof. Let ℓ < 1 be a constant such that any c-uniform K  To see the second inequality, note that the sum h∈E(H) E(Γ (c) (h)) ∩ R ′ counts each c-edge of R ′ at most |E(F )|−1 times. Thus, from the left-most and right-most expressions in the above inequality, we have So adding up (3) and (4), the proof is complete.