A note on the Tur\'an number of a Berge odd cycle

TL;DR

This paper improves upper bounds on the number of hyperedges in 3-uniform hypergraphs avoiding Berge odd cycles, extending previous results and providing new insights into Berge cliques.

Contribution

It presents tighter bounds on hyperedge counts for Berge odd cycles and introduces a more general theorem that enhances prior work.

Findings

Improved upper bounds for Berge odd cycle-free hypergraphs

New results on Berge cliques

Extension of previous bounds by F"uredi and "Ozkahya

Abstract

In this note we obtain upper bounds on the number of hyperedges in 3-uniform hypergraphs not containing a Berge cycle of given odd length. We improve the bound given by F\"uredi and \"Ozkahya in 2017. The result follows from a more general theorem. We also obtain some new results for Berge cliques.

Equations

$$ex(n,K_r,F)\le e{x}_r(n,\text{Berge-}F)\le ex(n,K_r,F)+ex(n,F)$$

$$e{x}_3(n,\text{Berge-}{C}_{2k+1})\le ex(n,K_3,C_{2k+1})+4ex(n,C_{2k})+12e{x}_3^{lin}(n,\text{Berge-}{C}_{2k+1}),$$

$$g_r(G)\le 2^{r-1}ex(n,F_0)-m+2cm/r\le \max \left\{1,\frac{2c}{r}\right\}(2^{r-1}ex(n,F_0)-m+m)=$$

$$\max \left\{1,\frac{2c}{r}\right\}{2}^{r-1}ex(n,F_0).$$

Full text

Keywords: Berge, hypergraph, cycle, Turán number

1 Introduction

We say that a hypergraph ${\mathcal{H}}$ is a Berge copy of a graph $F$ (in short: ${\mathcal{H}}$ is a Berge-$F$) if $V(F)\subset V({\mathcal{H}})$ and there is a bijection $f:E(F)\rightarrow E({\mathcal{H}})$ such that for any $e\in E(F)$ we have $e\subset f(e)$. This definition was introduced by Gerbner and Palmer [11], extending the well-established notion of Berge cycles and paths. Note that there are several non-uniform Berge copies of $F$, and a hypergraph ${\mathcal{H}}$ is a Berge copy of several graphs. A particular copy of $F$ defining a Berge-$F$ is called its core. Note that there can be multiple cores in a Berge-$F$.

We denote by $ex_{r}(n,\text{Berge-}F)$ the largest number of hyperedges in an $r$-uniform Berge-$F$-free hypergraph on $n$ vertices. There are several papers dealing with $ex_{r}(n,\text{Berge-}C_{k})$ (e.g. [8, 14, 15, 16]) or $ex_{r}(n,\text{Berge-}F)$ in general (e.g. [9, 10, 11, 12, 20]). For a short survey on this topic see Subsection 5.2.2 in [13].

In this note we consider $ex_{3}(n,\text{Berge-}C_{k})$. In the case $k=5$, this was first studied by Bollobás and Győri [2]. They showed $ex_{3}(n,\text{Berge-}C_{5})\leq\sqrt{2}n^{3/2}+4.5n$. This bound was improved to $(0.254+o(1))n^{3/2}$ by Ergemlidze, Győri and Methuku [5]. For cycles of any length, Győri and Lemons [15, 16] proved $ex_{r}(n,\text{Berge-}C_{k})=O(n^{1+1/\lfloor k/2\rfloor})$. The constant factors were improved by Jiang and Ma [18], and in the case $k$ is even by Gerbner, Methuku and Vizer [10]. In the 3-uniform case, Füredi and Özkahya [8] obtained better constant factors (depending on $k$). In the case $k$ is even, further improvements were obtained by Gerbner, Methuku and Vizer [10] and by Gerbner, Methuku and Palmer [9].

A closely related area is counting triangles in $C_{k}$-free graphs. More generally, let $ex(n,H,F)$ denote the maximum number of copies of $H$ in an $F$-free graph on $n$ vertices. After some sporadic results, the systematic study of these problems (often called generalized Turán problems) was initiated by Alon and Shikhelman [1]. Their connection to Berge hypergraphs was established by Gerbner and Palmer [12], who proved

[TABLE]

for any $r$, $n$ and $F$.

Counting triangles in $C_{k}$-free graphs and counting hyperedges in Berge-$C_{k}$-free 3-uniform hypergraphs was handled together already by Bollobás and Győri [2] for $C_{5}$, and by Füredi and Özkahya [8], who proved $ex(n,K_{3},C_{2k})\leq\frac{2k-3}{3}ex(n,C_{2k})$ and $ex_{3}(n,\text{Berge-}C_{2k})\leq\frac{2k}{3}ex(n,C_{2k})$. Their upper bound for $ex(n,K_{3},C_{2k})$ is still the best known bound, but their other upper bound was improved to $ex_{3}(n,\text{Berge-}C_{2k})\leq\frac{2k-3}{3}ex(n,C_{2k})$ by Gerbner, Methuku and Vizer [10] in the case $k\geq 5$ and by Gerbner, Methuku and Palmer [9] in the case $k=3,4$.

In the case of forbidden cycles of any odd length, the number of triangles was first studied by Győri and Li [17], who proved111We note that the bound is incorrectly stated in their paper [17]. $ex(n,K_{3},C_{2k+1})\leq\frac{(2k-2)(16k-1)}{3}ex(n,C_{2k})$. It was improved independently by Füredi and Özkahya [8] and by Alon and Shikhelman [1]. The latter had the stronger bound $ex(n,K_{3},C_{2k+1})\leq\frac{16(k-1)}{3}ex(\lceil n/2\rceil,C_{2k})$. In the case $k=2$, the current best bound $ex(n,K_{3},C_{5})\leq 0.231975n^{3/2}$ is due to Ergemlidze and Methuku [6].

Füredi and Özkahya [8] obtained the currently best upper bound on the Berge version by showing

[TABLE]

where $ex_{r}^{lin}(n,\text{Berge-}F)$ denotes the largest number of hyperedges in an $r$-uniform Berge-$F$-free linear hypergraph on $n$ vertices. Recall that a linear hypergraph is one in which any two hyperedges share at most one vertex.

In this note we improve the bound (1.1). Recall that we have $ex_{3}(n,\text{Berge-}C_{2k+1})\geq ex(n,K_{3},C_{2k+1})$, thus we cannot hope for a huge improvement, especially as $ex(n,K_{3},C_{2k+1})$ might be the largest of the three terms. Indeed, the best upper bound currently known is $O(n^{1+1/k})$ for all the three terms, but the dependence of the known upper bound in $k$ is the largest for $ex(n,K_{3},C_{2k+1})$ (we will state these bounds after Theorem 1.2).

Recall that in case of $C_{2k}$, the two upper bounds obtained by Füredi and Özkahya [8] were $ex(n,K_{3},C_{2k})\leq\frac{2k-3}{3}ex(n,C_{2k})$ and $ex_{3}(n,\text{Berge-}C_{2k})\leq\frac{2k}{3}ex(n,C_{2k})$, and the Berge bound was improved in [10, 9] to match the generalized Turán bound. Our goal would be to do the same here and get rid of the terms $4ex(n,C_{2k+1})+12ex_{3}^{lin}(n,\text{Berge-}C_{2k+1})$ in (1.1). We cannot achieve that, but we decrease these additional terms. Recall that the currently best bound for the generalized Turán problem is $ex(n,K_{3},C_{2k+1})\leq\frac{16(k-1)}{3}ex(\lceil n/2\rceil,C_{2k})$ by Alon and Shikhelman [1]. Our new upper bound on $ex_{3}(n,\text{Berge-}C_{2k+1})$ is larger than that bound by $ex_{3}^{lin}(n,\text{Berge-}C_{2k+1})$. We wonder if it is an example of a more general phenomenon and similar bounds could be obtained for other graphs.

The way we use the linearity involves subdividing an edge $uv$, i.e. deleting it and adding $uw$ and $vw$ for a new vertex $w$. Our method uses only the following two properties of $C_{2k+1}$: it can be obtained from $C_{2k}$ by subdividing an edge and deleting a vertex from $C_{2k+1}$ we obtain a path. In the next theorem we state our result in the most general form.

Theorem 1.1**.**

Let $F$ be a connected graph obtained from $F_{0}$ by subdividing an edge and $F^{\prime}$ be obtained from $F$ by deleting a vertex. Let $c=c(n)$ be such that $ex(n,K_{r-1},F^{\prime})\leq cn$ for every $n$. Then we have

(i) $ex_{r}(n,\text{Berge-}F)\leq ex(n,K_{r},F)+2^{r-1}ex(n,F_{0})+ex_{r}^{lin}(n,\text{Berge-}F)$,

(ii) $ex_{r}(n,\text{Berge-}F)\leq\max\left\{1,\frac{2c}{r}\right\}2^{r-1}ex(n,F_{0})+ex_{r}^{lin}(n,\text{Berge-}F)$.

In the case $F=C_{2k+1}$ we have $F_{0}=C_{2k}$ and $F^{\prime}=P_{2k}$, the path on $2k$ vertices. A theorem of Luo [19] shows $ex(n,K_{r-1},P_{2k})\leq\frac{n}{2k-1}\binom{2k-1}{r-1}$, but what we need for the 3-uniform case is the Erdős-Gallai theorem [4] showing $ex(n,P_{2k})\leq(k-1)n$. Using this, (ii) of Theorem 1.1 gives $ex_{3}(n,\text{Berge-}C_{2k+1})\leq\frac{8k-8}{3}ex(n,C_{2k})+ex_{3}^{lin}(n,\text{Berge-}C_{2k+1})$ if $k>2$. We can improve this a little bit.

Theorem 1.2**.**

If $k>2$, then $ex_{3}(n,\text{Berge-}C_{2k+1})\leq\frac{16k-16}{3}ex(\lceil n/2\rceil,C_{2k})+ex_{3}^{lin}(n,\text{Berge-}C_{2k+1})$

$\leq\left(\frac{1280k-1280}{3}\sqrt{k}\log k\right)\lceil n/2\rceil^{1+1/k}+2kn^{1+1/k}+9kn+\frac{16k-16}{3}10k^{2}\lceil n/2\rceil$.

The bound in Theorem 1.2 is currently stronger than the bound given by (i) of Theorem 1.1 for $F=C_{2k+1}$ and $r=3$. However, an improvement on $ex(n,K_{3},C_{2k+1})$ would immediately improve the bound in (i). Any significant improvement would make (i) stronger than Theorem 1.2 for $F=C_{2k+1}$.

The second inequality in Theorem 1.2 follows from known results. Füredi and Özkahya [8] proved $ex_{3}^{lin}(n,\text{Berge-}C_{2k+1})\leq 2kn^{1+1/k}+9kn$, and Bukh and Jiang [3] obtained the strongest bound on the Turán number of even cycles by showing $ex(n,C_{2k})\leq 80\sqrt{k}\log kn^{1+1/k}+10k^{2}n$. As we do not have good lower bounds on $ex(n,C_{2k})$, we cannot be sure that the first term is actually the larger term. However, if $ex_{3}^{lin}(n,\text{Berge-}C_{2k+1})$ is the larger term, then our improvement on the upper bound of $ex_{3}(n,\text{Berge-}C_{2k+1})$ is more significant, as we changed the constant factor of that term from 12 to 1. Obviously we have $ex_{3}^{lin}(n,\text{Berge-}C_{2k+1})\leq ex_{3}(n,\text{Berge-}C_{2k+1})$, hence further improvement is impossible here.

We prove Theorem 1.1 by combining the ideas of [8] and [1] with the methods developed in [9, 10]. In the next section we state some lemmas needed for the proof. We give a new proof of a lemma by Gerbner, Methuku and Palmer [9], and we strengthen the lemma a little bit. This strengthens results on $ex_{r}(n,\text{Berge-}K_{k})$ for some values of $r$, $k$ and $n$. In Section 3 we prove Theorems 1.1 and 1.2.

2 Lemmas

We say that a graph $G$ is red-blue if each of its edges is colored with one of the colors red and blue. For a red-blue graph $G$, we denote by $G_{red}$ the subgraph spanned by the red edges and $G_{blue}$ the subgraph spanned by the blue edges. For two graphs $H$ and $G$ we denote by $N(H,G)$ the number of subgraphs of $G$ that are isomorphic to $H$. Let $g_{r}(G)=|E(G_{red})|+N(K_{r},G_{blue})$.

Lemma 2.1** (Gerbner, Methuku, Palmer [9]).**

For any graph $F$ and integers $r$ and $n$, there is a red-blue $F$-free graph $G$ on $n$ vertices, such that $ex_{r}(n,\textup{Berge-}F)\leq g_{r}(G)$.

Note that an essentially equivalent version was obtained by Füredi, Kostochka and Luo [7]. The proof of Lemma 2.1 relies on a lemma about bipartite graphs (hidden in the proof of Lemma 2 in [9]). If $M$ is a matching and $ab$ is an edge in $M$, then with a slight abuse of notation we say $M(a)=b$ and $M(b)=a$.

Lemma 2.2**.**

Let $\Gamma$ be a finite bipartite graph with parts $A$ and $B$ and let $M$ be a largest matching in $\Gamma$. Let $B^{\prime}$ denote the set of vertices in $B$ that are incident to $M$. Then we can partition $A$ into $A_{1}$ and $A_{2}$ and partition $B^{\prime}$ into $B_{1}$ and $B_{2}$ such that for $a\in A_{1}$ we have $M(a)\in B_{1}$, and every neighbor of the vertices of $A_{2}$ is in $B_{2}$.

Here we present a proof that is built on the same principle, but is somewhat simpler than the proof found in [9]. Before that, let us recall the well-known notion of alternating paths. Given a bipartite graph $\Gamma$ and a matching $M$ in it, a path $P$ in $\Gamma$ is called alternating if its first edge is not in $M$, and then it alternates between edges in $M$ and edges not in $M$, finishing with an edge not in $M$. It is well-known and easy to see that deleting the edges of $P$ from $M$ and replacing them with the edges of $P$ that were not in $M$, we obtain another matching, that is larger than $M$.

Proof.

First we build a set $V^{\prime}\subset V(\Gamma)$ in the following way. Let $V_{0}$ be the set of vertices in $A$ that are not incident to any edges of $M$. Then in the first step we add to $V_{0}$ the set of vertices in $B$ that are neighbors of a vertex in $V_{0}$, to obtain $V_{1}$. In the second step we add to $V_{1}$ the vertices in $A$ that are connected to a vertex in $V_{1}$ by an edge in $M$, to obtain $V_{2}$. Similarly, in the $i$th step, if $i$ is odd we add to $V_{i-1}$ the set of vertices in $B$ that are neighbors of a vertex in $V_{i-1}$, while if $i$ is even, we add to $V_{i-1}$ the vertices in $A$ that are connected to a vertex in $V_{i-1}$ by an edge in $M$ (i.e. $M(b)$ for some $b\in B\cap V_{i-1}$), to obtain $V_{i}$. After finitely many steps, $V_{i}$ does not increase anymore, let $V^{\prime}$ be the resulting set of vertices.

We claim that no vertex from $B\setminus B^{\prime}$ can be in $V^{\prime}$. Indeed, such a vertex could be reached by an alternating path from a vertex in $A$ that is not incident to $M$, thus $M$ is not a largest matching, a contradiction.

Then let $A_{2}=A\cap V^{\prime}$, $A_{1}=A\setminus A_{2}$, $B_{2}=B^{\prime}\cap V^{\prime}$ and $B_{1}=B^{\prime}\setminus B_{2}$. A vertex in $A_{2}$ cannot be connected to a vertex $v$ not in $B_{2}$, as $v$ could be added to $V^{\prime}$ then. Similarly, for a vertex $u\in A_{1}$, $M(u)$ has to be in $B_{1}$, otherwise $M(u)$ is in $B_{2}$ and then $u$ can be added to $V^{\prime}$.

∎

Let us briefly describe how we can apply this lemma to obtain Lemma 2.1. We take a Berge-$F$-free $r$-uniform hypergraph ${\mathcal{H}}$ on $n$ vertices. Let $A$ be the set of hyperedges in ${\mathcal{H}}$ and $B$ be the set of sub-edges of these hyperedges (by edge and sub-edge we always mean an edge of size two, i.e. a pair of vertices). We connect $a\in A$ to $b\in B$ if $a\supset b$. Let $\Gamma$ denote this auxiliary bipartite graph. Let $M$ be an arbitrary largest matching and $B^{\prime}$ be the vertices of $B$ incident to the edges in $M$. It is easy to see that the elements of $B^{\prime}$ form an $F$-free graph which we call $G$. Indeed, otherwise $M$ defines the bijection between a copy of $F$ and hyperedges in ${\mathcal{H}}$ to form a Berge-$F$.

Now we apply Lemma 2.2 to $\Gamma$ and $M$. We define a red-blue coloring of $G$ by taking the edges of $G$ in $B_{1}$ to be the red edges, and the edges of $G$ in $B_{2}$ to be the blue edges. We have $|{\mathcal{H}}|=|A_{1}|+|A_{2}|=|B_{1}|+|A_{2}|=|E(G_{red})|+|A_{2}|$. As hyperedges in $A_{2}$ have all their neighbors in $B_{2}$, they each contain a blue $K_{r}$, which is distinct from the other blue $r$-cliques obtained this way, showing $|A_{2}|\leq N(K_{r},G_{blue})$.

Let us remark here that Lemma 2.2 also gives some information on the structure of $G$. If there is $a\in A_{1}$ that has a neighbor $b\in B\setminus B^{\prime}$, then we could obtain another matching $M^{\prime}$ by changing the neighbor of $a$ to $b$, i.e. $M^{\prime}(a)=b$ and if $a^{\prime}\neq a$, then $M^{\prime}(a^{\prime})=M(a^{\prime})$. Then $B^{\prime}$ is replaced by $B^{\prime\prime}=B^{\prime}\setminus\{M(a)\}\cup\{b\}$. In this case the same partition of $A$ into $A_{1}$ and $A_{2}$, and the partition of $B^{\prime\prime}$ into $B_{2}$ and $B^{\prime\prime}\setminus B_{2}$ satisfies Lemma 2.2. This means for $G$ that we can delete the (red) edge $M(a)$ and replace it with the edge $b$, to obtain another $F$-free graph.

If on the other hand the vertices in $A_{1}$ have all their neighbors in $B^{\prime}$, then we could recolor the red edges to blue. Therefore, in $G$ we can delete an edge and add another edge so that the resulting graph is still $F$-free. Let $\alpha=\alpha_{F,n}$ be the largest value of $g_{r}(G^{\prime})$, where $G^{\prime}$ is an $n$-vertex $F$-free blue-red graph. Assume that each $n$-vertex $F$-free blue-red graph $G^{\prime}$ with $g_{r}(G^{\prime})=\alpha$ is not monoblue and we cannot delete an edge and add another edge to $G^{\prime}$ so that the resulting graph is still $F$-free. Then by the above, $G$ cannot be one of these graphs, thus $ex_{r}(n,\textup{Berge-}F)\leq g_{r}(G)<\alpha$. This is usually a negligible improvement, as we often do not even know the order of magnitude.

However, if $F=K_{k}$, Gerbner, Methuku and Palmer [9] proved that $\alpha_{K_{k},n}=\max\{g_{r}(T_{B}(n,k-1)),g_{r}(T_{R}(n,k-1))\}$, where $T_{B}(n,k-1)$ is the monoblue Turán graph $T(n,k-1)$ and $T_{R}(n,k-1)$ is the monored Turán graph $T(n,k-1)$. We mention without going into the details that their proof also shows that for any other graphs $G$ we have $g_{r}(G)<\alpha_{K_{k},n}$. As we cannot delete an edge from $T(n,k-1)$ and add another edge to obtain a $K_{k}$-free graphs, we do have an improvement. For example, if $r=4$ and $k=5$, then the result in [9] determines $ex_{4}(n,\textup{Berge-}K_{k})$ for $n\geq 11$. For $n=10$, $T(10,4)$ has 36 copies of $K_{4}$ and 37 edges. Therefore, (as $ex(n,K_{r},F)$ is a lower bound on $ex_{r}(n,\textup{Berge-}F)$), we have $36\leq ex_{4}(n,\textup{Berge-}K_{k})\leq 37$. With our new observation, we know $ex_{4}(n,\textup{Berge-}K_{k})=36$.

3 Proof of Theorems 1.1 and 1.2

Let ${\mathcal{H}}$ be a Berge-$F$-free $r$-graph on $n$ vertices. We say that an edge $uv$ with $u,v\in V({\mathcal{H}})$ is $t$-heavy if $u,v$ are contained together in exactly $t$ hyperedges. First we will build a linear subhypergraph ${\mathcal{H}}_{1}$ in a greedy way: if we can find a hyperedge $H$ that does not share an edge with any hyperedge in ${\mathcal{H}}_{1}$, we add $H$ to ${\mathcal{H}}_{1}$, and then repeat this procedure. By definition, ${\mathcal{H}}_{1}$ is linear. Let ${\mathcal{H}}_{2}$ consist of the remaining hyperedges. Note that $|{\mathcal{H}}|=|{\mathcal{H}}_{1}|+|{\mathcal{H}}_{2}|\leq ex_{r}^{lin}(n,\text{Berge-}F)+|{\mathcal{H}}_{2}|$, and the remainder of the proof is for proving the needed upper bound on $|{\mathcal{H}}_{2}|$.

We build an auxiliary bipartite graph $\Gamma$ in the usual way: let $A$ be the set of hyperedges in ${\mathcal{H}}_{2}$ and $B$ be the set of sub-edges of these hyperedges. We connect $a\in A$ to $b\in B$ if $a\supset b$. We will let $M$ be a largest matching in $\Gamma$, however, we do not choose $M$ arbitrarily. Let $M_{0}$ be an arbitrary largest matching in $\Gamma$. Let $B^{\prime}$ be the set of vertices in $B$ that are incident to some edge of $M_{0}$ and $A_{0}$ denote the set of vertices in $A$ that are incident to some edge of $M_{0}$. Now a hyperedge $a\in A_{0}$ contains a sub-edge $M_{0}(a)$, at least one sub-edge $b_{0}$ shared with a hyperedge in ${\mathcal{H}}_{1}$, maybe some sub-edges that are matched to some other $a^{\prime}\in A$, and maybe some other sub-edges $b\in B\setminus B^{\prime}$. We have the option to replace in $M_{0}$ the edge between $a$ and $M_{0}(a)$ with any of the edges of $\Gamma$ between $a$ and an unused sub-edge of $a$, to obtain another largest matching. We will build a largest matching $M$, that contains the same vertices ($A_{0}$) from $A$ as $M_{0}$.

For $a\in A_{0}$, we pick $M(a)$ to be one of the sub-edges $b\in B$ of $a$ (potentially we let $M(a)=M_{0}(a)$) in the following way: $M(a)$ should share exactly one vertex with $b_{0}$ (where $b_{0}$ is a sub-edge that is also a sub-edge of a hyperedge in ${\mathcal{H}}_{1}$) if possible. We go through the hyperedges greedily; as long as there is a hyperedge $a\in A_{0}$ such that $M_{0}(a)$ can be changed in this way, we execute the change (it is possible that $M_{0}(a)$ cannot be changed originally, but later a sub-edge of $a$ that is $M_{0}(a^{\prime})$ becomes free to use, when $M(a^{\prime})$ is chosen to be different from $M_{0}(a^{\prime})$). This process finishes after finitely many (at most $|A_{0}|$) steps, as we change $M_{0}(a)$ to $M(a)$ at most once for every $a\in A_{0}$. After this, we rename the unchanged $M_{0}(a)$ to $M(a)$.

The resulting matching $M$ has the following property: for every $a\in A_{0}$, $a$ shares a sub-edge $b_{0}$ with a hyperedge in ${\mathcal{H}}_{1}$, such that that either $M(a)$ shares exactly one vertex with $b_{0}$, or all the sub-edges of $a$ sharing exactly one vertex with $b_{0}$ are $M(a^{\prime})$ for some $a^{\prime}\in A_{0}$.

Now we can apply Lemma 2.2 to $\Gamma$ and $M$ to obtain $A_{1},A_{2},B_{1},B_{2}$. Let us call the elements of $B_{1}$ red edges and the elements of $B_{2}$ blue edges. Let $G$ be the graph consisting of all the red and blue edges. Then $G$ is obviously $F$-free.

Let us now take a random partition of $V({\mathcal{H}})$ into $V_{1}$ and $V_{2}$. For every $a\in A_{0}$, we look at $b=M(a)$. If the two vertices of $b$ are in one part, and all the other vertices of $a$ are in the other part, we keep $a$, otherwise we delete it. Let $A^{*}$ denote the set of elements in $A$ that are not deleted (note that elements in $A\setminus A_{0}$ are never deleted, thus are in $A^{*}$). Let $G^{\prime}$ be the graph consisting of the elements of $B^{\prime}$ that are connected by an edge in $M$ to an element of $A^{*}$. Then $G^{\prime}$ is obviously $F$-free, as it is a subgraph of $G$.

Claim 3.1**.**

$G^{\prime}$* is $F_{0}$-free, where $F_{0}$ is any graph for which F can be obtained from $F_{0}$ by subdividing an edge of $F_{0}$.*

Proof.

Let us assume we are given a copy $Q$ of $F_{0}$ in $G^{\prime}$ such that $uv$ is the edge that needs to be subdivided to obtain $F$. Observe that there is no edge between $V_{1}$ and $V_{2}$ in $G^{\prime}$, thus $Q$ is in one of them, say $V_{1}$. Let $w$ be a vertex of $M(uv)$ with $u\neq w\neq v$, then $w\in V_{2}$, thus $w$ is not in $Q$.

We say that a hyperedge $H$ in ${\mathcal{H}}$ is good if $H$ contains $u$ and $w$ for some $w\in M(uv)\setminus\{u,v\}$ and $H$ is not $M(e)$ for any edge $e$ of $Q$. If there is a good hyperedge, then we build a Berge-$F$ with the following core: we subdivide $uv$ with $w$. For each edge $e$ of this core we assign $M(e)$ except for $uw$ (where we assign $H$) and $vw$ (where we assign $M(uv)$). This way we obtain a Berge-$F$, a contradiction.

$M(uv)$ shares at least one sub-edge with a hyperedge $H\in{\mathcal{H}}_{1}$. If the sub-edge shares exactly one vertex with $uv$, then $H$ is good and we are done. Thus every sub-edge of $M(uv)$ shared with a hyperedge in ${\mathcal{H}}_{1}$ has to contain none or both of $u$ and $v$. In both cases, when we tried to change $M_{0}(M(uv))$ when constructing $M$, we failed, because all such edges are matched to some other hyperedges of ${\mathcal{H}}_{2}$. In particular, $uw$ is $M(a)$ for some $a\in A_{0}$ and for some $w\in M(u,v)\setminus\{u,v\}$. Observe that $w$ is in $V_{2}$, thus $M(a)$ has vertices from both parts $V_{1}$ and $V_{2}$, hence $a$ cannot be in $A^{*}$ by the definition of $A^{*}$. This implies $a$ is good, finishing the proof.

∎

The above claim implies $G^{\prime}$ has at most $ex(n,F_{0})$ edges. For an arbitrary $a\in A$, the probability that $a$ is in $A^{*}$ is at least $1/2^{r-1}$. Let $S$ be any subset of $A$, then we have that the expected value of the number of hyperedges in $A^{*}\cap S$ is at least $|S|/2^{r-1}$, thus there is a partition with $|A^{*}\cap S|\geq|S|/2^{r-1}$.

There are $|B_{1}|=|A_{1}|$ red edges in $G$, and there is a random partition where at least $|A_{1}|/2^{r-1}$ elements of $A_{1}$ are undeleted, hence there are at least $|A_{1}|/2^{r-1}$ red edges in $G^{\prime}$. This implies $|A_{1}|/2^{r-1}\leq ex(n,F_{0})$. Hence there are at most $2^{r-1}ex(n,F_{0})$ red edges altogether. For the total number of edges in $G$ we can use the same argument: there is a random partition where at least $|A_{0}|/2^{r-1}$ hyperedges in $A_{0}$ are undeleted, thus for the $G^{\prime}$ defined by that partition, we have $|A_{0}|=|E(G)|\leq 2^{r-1}|E(G^{\prime})|\leq 2^{r-1}ex(n,F_{0})$.

Observe that we have $|{\mathcal{H}}_{2}|=|A_{1}|+|A_{2}|\leq|A_{1}|+N(K_{r},G_{blue})\leq|A_{1}|+ex(n,K_{r},F)$, hence we are done with the proof of (i).

Note that $G$ is not necessarily $F_{0}$-free, but it is $F$-free. Let $m$ be the number of blue edges in $G$, then $G$ has at most $2^{r-1}ex(n,F_{0})-m$ red edges. An argument of Gerbner, Methuku and Vizer [10] bounds the number of $r$-cliques in $F$-free graphs with the given number of vertices and edges. For sake of completeness, we include the argument here.

Let $d(v)$ be the degree of $v$ in $G_{\textup{blue}}$. Obviously the neighborhood of every vertex in $G_{\textup{blue}}$ is $F^{\prime}$-free. An $F^{\prime}$-free graph on $d(v)$ vertices contains at most $ex(d(v),K_{r-1},F^{\prime})\leq cd(v)$ copies of $K_{r-1}$. Thus $v$ is contained in at most $cd(v)$ copies of $K_{r}$ in $G_{\textup{blue}}$. If we sum, for each vertex, the number of $K_{r}$’s containing a vertex, then each $K_{r}$ is counted $r$ times. On the other hand as $\sum_{v\in V(G_{\textup{blue}})}d(v)=2|E(G_{blue})|=2m$, we have $\sum_{v\in V(G_{\textup{blue}})}cd(v)=2cm$. This gives that the number of blue $K_{r}$’s is at most $2cm/r$. Thus we have

[TABLE]

The above inequality, together with Lemma 2.1 implies that $|{\mathcal{H}}_{2}|\leq\max\left\{1,\frac{2c}{r}\right\}2^{r-1}ex(n,F_{0})$, finishing the proof of (ii).

Now we show how to obtain the small improvement needed to prove Theorem 1.2. It is based on the proof of the upper bound on $ex(n,K_{3},C_{2k+1})$ in [1]. If $n$ is odd, replace it by $n+1$. As the stated upper bound is the same in both cases, obvious mononicity conditions show we can do this. Thus we can assume $n$ is even. When we take the random partition into $V_{1}$ and $V_{2}$, first we take a random partition into $n/2$ sets $U_{1},\dots,U_{n/2}$ of size 2, and then randomly put one vertex into $V_{1}$ and the other into $V_{2}$. The obtained graph $G^{\prime}$ will be $C_{2k}$-free, and it is divided into two components, hence it has at most $ex(|V_{1}|,C_{2k})+ex(|V_{2}|,C_{2k})$ edges. The way we chose $V_{1}$ ensures the above sum is $2ex(\lceil n/2\rceil,C_{2k})$. Then we can go through every step of the remaining part of the proof to obtain the result we need, if for an arbitrary $a\in A$, the probability that $a$ is in $A^{*}$ is still at least $1/2^{r-1}=1/4$. We will separate into cases according to the intersection of $a$ with the parts $U_{i}$. In case the three vertices of $a$ are in three different $U_{i}$’s, the probability is $1/4$. In case $a$ contains $U_{i}$ for some $i$, there are two cases. If $M(a)=U_{i}$, then the probability is [math], otherwise it is $1/2$. As $M(a)=U_{i}$ happens with probability $1/3$ (having the condition that $a$ contains $U_{i}$), for every $i$ we have that the probability of $a$ being in $A^{*}$ if $a$ contains $U_{i}$ is $\frac{2}{3}\cdot\frac{1}{2}\geq 1/4$.

This gives the first inequality of Theorem 1.2. As we have mentioned after the statement, the second inequality follows from earlier results, stated there.

Acknowledgements: Research supported by the National Research, Development and Innovation Office - NKFIH under the grants SNN 129364, KH 130371 and K 116769 and by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences.