Forbidding $K_{2,t}$ traces in triple systems

Let $H$ and $F$ be hypergraphs. We say $H$ contains $F$ as a trace if there exists some set $S \subseteq V(H)$ such that $H|_S:=\{E\cap S: E \in E(H)\}$ contains a subhypergraph isomorphic to $F$. In this paper we give an upper bound on the number of edges in a $3$-uniform hypergraph that does not contain $K_{2,t}$ as a trace when $t$ is large. In particular, we show that $ \lim_{t\to \infty}\lim_{n\to \infty} \frac{\mathrm{ex}(n, \mathrm{Tr}_3(K_{2,t}))}{t^{3/2}n^{3/2}} = \frac{1}{6}.$ Moreover, we show $\frac{1}{2} n^{3/2} + o(n^{3/2}) \leq \mathrm{ex}(n, \mathrm{Tr}_3(C_4)) \leq \frac{5}{6} n^{3/2} + o(n^{3/2})$.


Introduction
A hypergraph H is a family of subsets of some fixed ground set. The subsets are called the edges of H and the ground set is called the vertex set of H. We denote these sets by E(H) and V (H) respectively. If each edge of H contains exactly r elements, then we say that H is r-uniform.
A cornerstone of extremal combinatorics is the Turán problem. Broadly speaking, the Turán problem asks to determine the maximum number of edges in a hypergraph which contains no subhypergraphs isomorphic to a member of some given forbidden family.
In this paper, we study uniform hypergraphs with forbidden traces. Definition 1.1. Let F and T be uniform hypergraphs (possibly with different uniformities) with V (F ) ⊆ V (T ). We say that T is a trace of F on V (F ), or simply an F -trace, if there exists a bijection φ : E(F ) → E(T ) such that for every edge e ∈ E(F ), φ(e) ∩ V (F ) = e. We say a hypergraph H contains F as a trace if it contains a subhypergraph isomorphic to a trace of F .
Equivalently, H containing F as a trace means that there exists some set S of vertices (corresponding to V (F )) such that H| S := {E ∩ S : E ∈ E(H)} has a subhypergraph isormorphic to F . We note that in different contexts, traces are also called configurations [24] and induced Berge F 's [13].
For r ≥ 2, let Tr r (F ) denote the set of all r-uniform hypergraphs that are traces of F up to isomorphism. If F is a family of r-uniform hypergraphs, then the function ex(n, F) denotes the maximum number of edges in an n-vertex, r-uniform hypergraph with no subhypergraph isomorphic to a member of F. In particular, ex(n, Tr r (F )) is the maximum size of a hypergraph that does not contain F as a trace.
Forbidding traces in hypergraphs is closely related to the well known Berge Turán problem. Definition 1.2. Given hypergraphs F and T , we say T is a Berge F if there exists a bijection φ : E(F ) → E(T ) such that for every edge e ∈ E(F ), e ⊆ φ(e).
Let B r (F ) denote the set of all r-uniform hypergraphs that are Berge F 's up to isomorphism. Observe that Tr r (F ) ⊆ B r (F ). Consequently, ex(n, B r (F )) ≤ ex(n, Tr r (F )). (1) In this paper, we focus only on the case where F is a graph, particularly F = K 2,t .

Known extremal results for degenerate graphs
Generalizing a result of Mantel [22], Turán [25] determined ex(n, K t ), the maximum number of edges in an n-vertex graph without a copy of K t , for all t. Later results by Erdős, Stone, and Simonovits [7,8] established the asymptotic value of ex(n, F ) for any graph F which is nonbipartite.
In [16], Gerbner, Methuku, and Vizer proved an upper bound for the Turán number of Berge K 2,t in r-uniform hypergraphs. They obtained asymptotically sharp bounds for 3-uniform hypergraphs when t ≥ 7. This was later extended by Gerbner, Methuku, and Palmer [15] for t ≥ 4.

Known results for forbidden traces
Some earlier results for forbidding traces of graphs in hypergraphs where due to Mubayi and Zhao [23] who determined the asymptotic value of ex(n, Tr r (K s )) for all r when s ∈ {3, 4}. They also conjectured that for s ≥ 5, ex(n, Tr r (K s )) ∼ n s−1 s−1 .
Sali and Spiro [24] determined the order of magnitude of ex(n, Tr r (K s,t )) when t ≥ (s − 1)! + 1, s ≥ 2r − 4. Later, Füredi and Luo [13] generalized their proof to deduce the order of magnitude of ex(n, Tr r (F )) for all graphs F in terms of their generalized Turán numbers. In particular, they showed ex(n, Tr r (F )) = Θ( max 2≤s≤r ex(n, K s , F )), where ex(n, K s , F ) denotes another extremal function, the maximum number of copies of K s in an F -free graph on n vertices.
When F is non-bipartite, this implies ex(n, Tr r (F )) = Ω(n 2 ) for all r. This contrasts with the problem of forbidding Berge copies of F : Grosz, Methuku, and Tompkins [18] proved that for all F there exists an r 0 such that for all r ≥ r 0 , ex(n, B r (F )) = o(n 2 ). In particular, for large r, ex(n, B r (F )) = o(ex(n, F )).
For F = C 4 this gives the bounds

New Results
Our main result is an upper bound for ex(n, Tr 3 (K 2,t )) which is effective for large t.
Here and throughout log denotes the natural logarithm.
We note that the constant 55 can be improved by a more careful analysis. On the other hand, for t ≥ 4 we have ex(n, Tr r (K 2,t )) ≥ 1 6 (t − 1) 3/2 n 3/2 + o(n 3/2 ) by (1) and Theorem 1.4. This together with Theorem 2.1 gives the following.
Separately analysing the case for K 2,2 = C 4 , we obtain tighter bounds which significantly improves (2).

Main Lemmas and the Proof of Theorem 2.1
Given a hypergraph H, we define d H (x, y) to be the number of edges of H containing {x, y}, and we call this number the co-degree of {x, y}. We will often identify hypergraphs by their set of edges and write e.g. H \ A to denote the hypergraph H after deleting some set of edges A from E(H).
For a hypergraph H and δ ∈ R + , define That is, every edge in H − δ contains a pair with co-degree at most δ.
Let H be some 3-uniform hypergraph and fix δ ≥ 2. We partition the edges of H into sets with small, medium, and large co-degrees in the following manner: A is the set of edges containing at least one pair with co-degree 1; , B δ is the set of edges in which every pair has co-degree at least 2, but at least one pair of co-degree at most δ in H; and , C δ is the set of edges in which every pair is contained in at least δ other edges of H.
The bulk of the work in showing Theorem 2.1 will be in proving the following technical lemmas. For ease of notation, for δ ≥ 2 we define and when δ is clear from context we simply write ε.
If δ ≥ 14, then for any pair {x, y} we have Assuming these lemmas, we can prove the following technical theorem.
Theorem 3.5. Fix t and let g(t) be any function such that Let us first show that this implies our main result.
Proof of Theorem 2.1, assuming Theorem 3.5. Take g(t) = 1 7 t log(t), and note that t ≥ t/g(t) = 7 t/ log(t) ≥ 14 when t ≥ 14, so we can apply the bound of Theorem 3.5.
We also have Combining this with the inequality above gives the desired result.
It remains to prove Theorem 3.5.
Proof of Theorem 3.5, assuming Lemmas 3.1 -3.4. Let G be a graph on [n] whose edge set is obtained by selecting from each e ∈ A a pair of vertices with co-degree 1. Suppose there exists a subgraph K ⊆ G which is a copy of K 2,t . For every edge xy in K, there exists some vertex z such that {x, y, z} ∈ A. Note that we can not have z ∈ V (K), as if say xz ∈ E(K), then this implies there exists some other edge {x, z, w} ∈ A and hence d H (x, z) > 1, a contradiction to how A was defined. Therefore the edges of A corresponding to K in G intersect V (K) in exactly the edges of K. This forms a K 2,t trace in H, a contradiction. We conclude by Theorem 1.3 that
The rest of the paper is organized as follows. In Section 4 we introduce the notion of dominated sets and prove Lemmas 3.1-3.3. We prove Lemma 3.4 in Section 5. Finally, focusing on the t = 2 case, we prove Theorem 2.3 in Sections 6.
We gather some standard notation we use throughout the paper. For a graph G we let ∆(G) denote its maximum degree, and we define α(G) to be the size of a maximum independent set in G. N G (v) denotes the neighborhood set of v in G. We will write edges either as {x, y} or xy depending on the context, and similarly for hyperedges we write either {x, y, z} or xyz.

Dominated Sets and Co-Degrees
In the literature, a set A of vertices in a graph is a dominating set if every vertex is either in A or has a neighbor in A. Here we introduce the notion of dominated sets in graphs with loops. Suppose G is a graph, possibly with loops. We say D ⊆ V (G) is a dominated set if for every v ∈ D, either v has a loop edge or v has a neighbor outside of D. We define the degree of a vertex v in such a graph to be the number of edges incident to v (counting loops with multiplicity).  The other important observation we make is Indeed, every edge {x, u, v} ∈ E(H) contributes to an edge involving u in L x (possibly as a loop) unless v = y.
Thus our goal is to find large sets that are simultaneously dominated in two graphs. The most general lemma we have in this direction is the following, which is an easy adaptation of a standard proof for finding a small dominating set (see for example [2]). Recall that we define ε = ε δ = 1+log(δ+1) δ+1 . Lemma 4.2. Let G be an n-vertex graph with loops with minimum degree at least δ ≥ 2. Then G has a dominated set of size at least (1 − ε)n.
Proof. Let D ⊆ V (G) be a random set obtained by picking each vertex of G independently with probability p. Let T ⊆ D be the set of vertices of D which do not have loops and do not have neighbors outside of D. Observe that D \ T is a dominated set.
Any given v ∈ V (G) is in T with probability 0 if it has a loop and otherwise with probability at most p δ+1 , as all its neighbors and itself must be selected. Thus by linearity of expectation we have Taking p = 1 − log(δ + 1)/(δ + 1) gives a set of size at least n − 1+log(δ+1) δ+1 n.
This quickly gives an upper bound for the co-degrees of C δ .
Proof of Lemma 3.3. Assume we have some pair of vertices {x, y} and a set S of size at least (1 + 4ε)t such that {x, y, u} ∈ C δ for all u ∈ S. By definition of C δ , this implies that d H (x, u), d H (y, u) > δ for all u ∈ S. By (6) and Lemma 4.2, we can find sets D x , D y ⊆ S which are dominated in L x , L y of size at least (1 − ε)(1 + 4ε)t, and in particular D := D x ∩ D y will be dominated in both and have size at least (1 − 2ε)(1 + 4ε)t ≥ t, where we use that ε ≤ 1/4 whenever δ ≥ 14. Then H contains a K 2,t trace by Lemma 4.1, a contradiction.
We next want to prove a bound when L x , L y are only known to have minimum degree at least 1. We first need the following simple result, where we recall that G ⊆ G is called a spanning subgraph if V (G ) = V (G). Proof. We greedily build our subgraph. Suppose at step i, we have a subgraph with components S 1 , . . . , S i−1 such that each component is either a star or a vertex with a loop. Let V i−1 be the set of vertices covered by S 1 , . . . , S i−1 , and suppose there exists v ∈ V (G) \ V i−1 . If v has a loop, we set S i = {v}. Otherwise, let S i be the star with center v and leaf vertices N (v) \ V i−1 . Then S i has at least 1 leaf unless N G (v) \ V i−1 is empty. In this case, for any u ∈ N G (v) we have u ∈ S j for some j ≤ i − 1. If S j = {u}, that is, u has a loop, remove S j and let S i = vu. So suppose S j is a star. Note that u is not the center, otherwise v would also be in S j . If S j has at least two leaves, then we replace it with the star S j \ {u} and let S i = vu. Otherwise S j is a single edge, say wu. Then we remove S j and let S i be the star with edges uv, uw.
With this we can prove the following. Let G x , G y be graphs on S with minimum degree at least 1. Then there exists a set D which is dominated in both G x and G y of size at least |S|/3. Moreover, if |S| = 3, then one can find such a set with |D| = 2 unless G x ∪ G y is a K 3 (possibly with loops).
Proof. Let G x ⊆ G x , G y ⊆ G y be the subgraphs guaranteed by Lemma 4.3. Let C x consist of the centers of stars of order at least 2 in G x , where a center of a star of order 2 is chosen arbitrarily. Similarly define C y and set D = S \ (C x ∪ C y ). Note that by assumption on G x , every u ∈ D ⊆ S \ C x either has a loop or is adjacent to something in C x . The same holds for G y , so D is dominated in both graphs and hence also in G x , G y .
It remains to bound the size of D. Observe that every u ∈ D is adjacent to at most one vertex in each of G x , G y (namely the center of the star it's in). Thus for each vertex added to D we omitted at most two vertices from D, giving the first bound. If, say, S = {u, v, w} and uv / ∈ G x ∪ G y , then by the minimum degree conditions, each of u, v must be adjacent to either w or have a loop in G x , G y . Thus D = {u, v} is a dominated set.
We now prove Lemmas 3.1 and 3.2.
Proof of Lemma 3.1. Assume there exists a set S of size at least 3t − 2 and a pair {x, y} such that {x, y, u} ∈ E(H) \ A for all u ∈ S. By definition of A, this implies that d H (x, u), d H (y, u) ≥ 2 for all u ∈ S. Thus L x = L x (H, S, y) and L y = L y (H, S, x) have minimum degree at least 1 by (6), so by Lemma 4.4 we can find a set D ⊆ S which is simultaneously dominated in L x , L y of size at least |S|/3 ≥ t. Then H contains a K 2,t trace by Lemma 4.1, a contradiction.
For t = 2, if there exists such an S = {u, v, w}, then by Lemma 4.4 we can assume L x ∪ L y is a K 3 , and without loss of generality we can assume uv, uw ∈ L x . By definition this implies that {x, u, v}, {x, u, w} ∈ E(H). By definition of S there exist edges {x, y, v}, {x, y, w} ∈ E(H). These four edges form a C 4 trace on {y, u, v, w}, which is a contradiction.
Proof of Lemma 3.2. Recall that B δ is the set of edges of H \ A that contain a pair with co-degree at most δ. Let G be the graph on [n] whose edge set is obtained from B δ by taking from each e ∈ B δ a pair of vertices with co-degree at most δ in H \ A.
Observe that e(B δ ) ≤ δ · e(G) as each edge in G is mapped to by at most δ edges of H. We claim that G is K 2,r -free with r = k + 3t − 2, which will give the stated bound by Theorem 1.3. Indeed, assume for contradiction that G contained such a K 2,r on {x, y} ∪ {u 1 , . . . , u r }. Let S be the set of vertices u i of this K 2,r for which {x, y, u i } / ∈ E(H), and by assumption there are at least r − k = 3t − 2 such vertices. By definition of S, every vertex in L x , L y has degree at least 1, so by Lemma 4.4 we can find a set of size at least t that is dominated in L x , L y , giving a K 2,t trace by Lemma 4.1 which is a contradiction.

Proof of Lemma 3.4
Our proof of Lemma 3.4 involving hypergraphs with co-degrees at most k will be an adaptation of a proof in [16] concerning linear hypergraphs. Throughout this section, unless stated otherwise we will assume to be working in C δ , which we recall is the set of edges in H in which every pair has co-degree greater than δ in H. For ease of notation we let d(v) = d C δ (v). We now begin the formal proof.
For v ∈ V (C δ ), define the 1 and 2-neighborhood of v as First observe that if E is a set of edges containing some vertex v and V is the set of vertices u = v with u ∈ e for some e ∈ E, then as each vertex in V is contained in at most k edges with v.
Proof. Assume this was not the case for some x, y. Note that at most k of these edges contain x since {x, y} has co-degree at most k, so there exists a set of 1 2 k · 50t of these edges E which do not contain v. Let S = e∈E e \ {y}, and by (7) we have that |S| ≥ 50t.
In the language of the previous section, we define L x = L x (H, S, y) and L y = L y (H, S, x). By definition of C δ and (6) these graphs have minimum degree at least δ ≥ 14. By Lemma 4.2 we can find dominated sets D x , D y of size at least (1 − δ )|S| ≥ .51|S|, and thus D = D x ∩ D y is a set dominated in both L x , L y of size at least .02|S| ≥ t. This implies that H contains a K 2,t trace by Lemma 4.1, a contradiction.
We point out that the above bound can be further optimized, however such improvements will not affect our asymptotic result.
From now on we fix some v ∈ V (C δ ). For u ∈ N 1 (v), define Lemma 5.2.
Proof. Suppose for contradiction that |V u | > ((1 + 4 )t − 1)n. By the pigeonhole principle, there exists a vertex x / ∈ N 1 (v) and a set S ⊆ N 1 (v) of size at least (1 + 4ε)t such that x ∈ V u for all u ∈ S. Define L v = L v (H, S, x) and L x = L x (H, S, v). By assumption every u ∈ S is contained in an edge {u, v, w v }, {u, x, w x } ∈ E(C δ ), so by (6) these graphs have minimum degree at least δ. By Lemma 4.2 we can find a set D which is dominated in both of these graphs with size at least (1 − 2ε)(1 + 4ε)t ≥ t for δ ≥ 14. By Lemma 4.1 we conclude that H contains a K 2,t trace, a contradiction.
and by (7) we have |V u | ≥ 2 k |E u |, therefore By Lemma 5.2 and (7), Let d = 3e(C δ )/n denote the average degree of C δ . Then summing over the above inequality gives On the other hand, because |N 1 (u)| ≥ 2 k d(u) by (7), and because u ∈ N 1 (v) if and only if v ∈ N 1 (u), we can reverse the sum to get with the last step following from the Cauchy-Schwarz inequality. By combining (9) and (10), we find with b := 13 2 k 3 t and c := k 2 4 (1 + 4ε) t that where this last step used k ≥ (1 + 4ε)t. Thus giving the desired bound.

Proof of Theorem 2.3
In this section we refine our methods and prove Theorem 2.3 for forbidden C 4 traces. As many ideas are carried over from the proof of Theorem 3.5, we omit some of the redundant details. We note that the lower bound of Theorem 2.3 follows from Theorem 1.5, so it remains to prove the upper bound.
Let H be an n-vertex, 3-uniform hypergraph with no C 4 trace. Let A = H − 1 , i.e., the edges with at least one pair of co-degree 1, and B = H \ A. Let G A be a graph on [n] whose edge set is obtained by adding a pair of co-degree 1 from every edge of A. Then G A is C 4 -free and we have It remains to show that |B| ≤ Proof. Suppose there exists x, y ∈ V (H) and some set {u 1 , . . . , u 8 } ⊆ N 1 (x) ∩ N 1 (y). At most two u i 's, say u 7 and u 8 , are in edges of the form {x, y, u i } ∈ B. Let G be a graph on [6] where ij ∈ E(G) if and only if either {x, u i , u j } ∈ B or {y, u i , u j } ∈ B. Because pairs in B have co-degree at most 2, we have ∆(G) ≤ 4. In particular, there exists a non-adjacent pair, say {1, 2}. Let e x,1 be any edge of B containing {x, u 1 }, and note that y, u 2 / ∈ e x,1 . Similarly define e x,2 , e y,1 , e y 2 . Then these four edges form a C 4 trace in B, which is a contradiction. Now fix any vertex v ∈ V (H). As before, define E u = {e ∈ B : e ∩ N 1 (v) = {u}} and V u = {w ∈ N 2 (v) : ∃e ∈ E u , w ∈ e}. Since V u ⊆ N 1 (u) for all u, we have the following corollaries.
Proof. Suppose that there exists edges ua, wb ∈ E(G v ) such that ua, wb = uw. By the definition of G v , vua, vwb ∈ B. Note that a, b = x, since x / ∈ N 1 (v). Let e u ∈ E u and e w ∈ E w be edges containing x. Then the edges {vua, e u , e w , vwb} form a C 4 trace.
We define the following sets V u ⊆ V u for u ∈ N 1 (v). If d Gv (u) = 2, then V u = V u . Otherwise if N Gv (u) = {w}, set V u = V u \ V w .
By Corollary 6.2, |V u | ≥ |V u | − 7 for all u. By Claim 6.5, the V u sets are pairwise disjoint from each other. Therefore u∈N 1 (v) |V u | ≤ n and where the last inequality comes from the fact that |N 1 (v)| ≤ 2d(v).
Let d = 3e(H)/n denote the average degree of H. We have

Concluding remarks
It remains to determine the exact value of ex(n, Tr r (C 4 )), especially in the case where r ≥ 4. The current best upper bound is that given by Theorem 1.5. In particular, we know ex(n, Tr r (C 4 )) = Θ(n 3/2 ) for all r, but determining the limit (if it exists) lim n→∞ ex(n, Tr r (C 4 ))/n 3/2 is likely difficult, though not as difficult as the more general ex(n, B r (C 4 )) problem. For this problem, it is not even known if ex(n, B r (C 4 )) = Θ(n 3/2 ) for r large.