Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. Denote by $\ex^{\mathrm{conn}}_r(n,\mathcal{F})$ the maximum number of hyperedges in an $n$-vertex connected $r$-uniform hypergraph which contains no member of $\mathcal{F}$ as a subhypergraph. Denote by $\mathcal{B}C_k$ the Berge cycle of length $k$, and by $\mathcal{B}P_k$ the Berge path of length $k$. Füredi, Kostochka and Luo, and independently Győri, Salia and Zamora determined $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ provided $k$ is large enough compared to $r$ and $n$ is sufficiently large. For the case $k\le r$, Kostochka and Luo obtained an upper bound for $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$. In this paper, we continue investigating the case $k\le r$. We precisely determine $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ when $n$ is sufficiently large and $n$ is not a multiple of~$r$. For the case $k=r+1$, we determine $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ asymptotically.
Fix a hypergraph $\mathcal{F}$. A hypergraph $\mathcal{H}$ is called a {\it Berge copy of $\mathcal{F}$} or {\it Berge-$\mathcal{F}$} if we can choose a subset of each hyperedge of $\mathcal{H}$ to obtain a copy of $\mathcal{F}$. A hypergraph $\mathcal{H}$ is {\it Berge-$\mathcal{F}$-free} if it does not contain a subhypergraph which is Berge copy of $\mathcal{F}$. This is a generalization of the usual, graph based Berge hypergraphs, where $\mathcal{F}$ is a graph. In this paper, we study extremal properties of hypergraph based Berge hypergraphs and generalize several results from the graph ba…
For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be replacing each edge of $F$ with a hyperedge containing it. We say a hypergraph is Berge-$F$-saturated if it does not contain a Berge-$F$, but adding any hyperedge creates a copy of Berge-$F$. The $k$-uniform saturation number of Berge-$F$, $\mathrm{sat}_k(n,\text{Berge-}F)$ is the fewest number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show that $\mathrm{sat}_k(n,\text{Berge-}F) = O(n)$ for all graphs $F$ and uniformities $3\leq k\leq 5$, partially answering a conjecture o…
Given a graph $F$, a hypergraph is a Berge-$F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subgraph that is a Berge-$F$, but for any edge $e\in E(\overline{H})$, $H+e$ does. The $k$-uniform saturation number of Berge-$F$ is the minimum number of edges in a $k$-uniform Berge-$F$-saturated hypergraph on $n$ vertices. For $k=2$ this definition coincides with the classical definition of saturation for graphs. In this paper we study the saturation numbers for Berge triangles, paths, cycles, stars…
Given a graph $G$, we say a $k$-uniform hypergraph $H$ on the same vertex set contains a Berge-$G$ if there exists an injection $ϕ:E(G)\to E(H)$ such that $e\subseteqϕ(e)$ for each edge $e\in E(G)$. A hypergraph $H$ is Berge-$G$-saturated if $H$ does not contain a Berge-$G$, but adding any edge to $H$ creates a Berge-$G$. The saturation number for Berge-$G$, denoted $\mathrm{sat}_k(n,\text{Berge-}G)$ is the least number of edges in a $k$-uniform hypergraph that is Berge-$G$-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we al…
Given a set $R$, a hypergraph is $R$-uniform if the size of every hyperedge belongs to $R$. A hypergraph $\mathcal{H}$ is called \textit{covering} if every vertex pair is contained in some hyperedge in $\mathcal{H}$. In this note, we show that every covering $[3]$-uniform hypergraph on $n\geq 6$ vertices contains a Berge cycle $C_s$ for any $3\leq s\leq n$. As an application, we determine the maximum Lagrangian of $k$-uniform Berge-$C_{t}$-free hypergraphs and Berge-$P_{t}$-free hypergraphs.
We consider two extremal problems for set systems without long Berge cycles. First we give Dirac-type minimum degree conditions that force long Berge cycles. Next we give an upper bound for the number of hyperedges in a hypergraph with bounded circumference. Both results are best possible in infinitely many cases.
For a graph G, a hypergraph H is called Berge-G if there is a hypergraph H', isomorphic to H, containing all vertices of G, so that e is contained in f(e) for each edge e of G, where f is a bijection between E(G) and E(H'). The set of all Berge-G hypergraphs is denoted B(G). A hypergraph H is called Berge-G saturated if it does not contain any subhypergraph from B(G), but adding any new hyperedge of size at least 2 to H creates such a subhypergraph. Each Berge-G saturated hypergraph has at least |E(G)|-1 hyperedges. We show that for each graph G that is not a certain star and for any n at leas…
In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph $\mathcal{H}$ is a $t$-heavy copy of a graph $F$ if there is a copy of $F$ on its vertex set such that each edge of $F$ is contained in at least $t$ hyperedges of $\mathcal{H}$. $\mathcal{H}$ is a $t$-wise Berge copy of $F$ if additionally for distinct edges of $F$ those $t$ hyperedges are distinct. We extend known upper bounds on the Turán number of Be…
Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an $n$-vertex $3$-uniform linear hypergraph, without a Berge path of length $k$ as a subgraph is at most $\frac{(k-1)}{6}n$ for $k\geq 4$.
In this paper, we consider maximum possible value for the sum of cardinalities of hyperedges of a hypergraph without a Berge $4$-cycle. We significantly improve the previous upper bound provided by Gerbner and Palmer. Furthermore, we provide a construction that slightly improves the previous lower bound.
We generalize a result of Balister, Gy{ő}ri, Lehel and Schelp for hypergraphs. We determine the unique extremal structure of an $n$-vertex, $r$-uniform, connected, hypergraph with the maximum number of hyperedges, without a $k$-Berge-path, where $n \geq N_{k,r}$, $k\geq 2r+13>17$.
We study the structure of $r$-uniform hypergraphs containing no Berge cycles of length at least $k$ for $k \leq r$, and determine that such hypergraphs have some special substructure. In particular we determine the extremal number of such hypergraphs, giving an affirmative answer to the conjectured value when $k=r$ and giving a a simple solution to a recent result of Kostochka-Luo when $k < r$.
A hypergraph $H=(V(H), E(H))$ is a Berge copy of a graph $F$, if $V(F)\subset V(H)$ and there is a bijection $f:E(F)\rightarrow E(H)$ such that for any $e\in E(F)$ we have $e\subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain any Berge copies of $F$. We address the saturation problem concerning Berge-$F$-free hypergraphs, i.e., what is the minimum number $sat_r(n,F)$ of hyperedges in an $r$-uniform Berge-$F$-free hypergraph $H$ with the property that adding any new hyperedge to $H$ creates a Berge copy of $F$. We prove that $sat_r(n,F)$ grows linearly in $n$ if $F$ is either c…
For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$. We denote the maximum number of hyperedges in an $n$-vertex $r$-uniform Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F).$ In this paper we prove two general lemmas concerning the maximum size of a Berge-$F$-free hypergraph and use them to establish new results and improve several old results. I…
Let $G$ be a graph and $\mathcal{H}$ be a hypergraph both on the same vertex set. We say that a hypergraph $\mathcal{H}$ is a \emph{Berge}-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for $e \in E(G)$ we have $e \subset f(e)$. This generalizes the established definitions of "Berge path" and "Berge cycle" to general graphs. For a fixed graph $G$ we examine the maximum possible size (i.e.\ the sum of the cardinality of each edge) of a hypergraph with no Berge-$G$ as a subhypergraph. In the present paper we prove general bounds for this maximum when $G$ is an arbitr…
Recently, Berge theta hypergraphs have received special attention due to the similarity with Berge even cycles. Let $r$-uniform Berge theta hypergraph $Θ_{\ell,t}^{B}$ be the $r$-uniform hypergraph consisting of $t$ internally disjoint Berge paths of length $\ell$ with the same pair of endpoints. In this work, we determine the Turán number of $3$-uniform Berge theta hypergraph when $\ell=3$ and $t$ is relatively small. More precisely, we provide an explicit construction giving \begin{align*} \textup{ex}_{3}(n,Θ_{3,217}^{B})=Ω(n^{\frac{4}{3}}). \end{align*} This matches an earlier upper bound b…
Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $ϕ: E(G)\to E(H)$ such that $e\subseteq ϕ(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the $p$-spectral radius $λ^{(p)}(H)$ of $H$ is defined as \[ λ^{(p)}(H):=\max_{{\bf x}\in\mathbb{R}^n,\,\|{\bf x}\|_p=1} r\sum_{\{i_1,i_2,\ldots,i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r}. \] In this paper, we study the $p$-spectral radius of Berge-$G$ hypergraphs. We determine the $3$-uniform hypergraphs with maximum $p$-spectral radius for $p\geq 1$ among Berge-$G$ hypergraph…
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 i…
Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The maximum number of edges in an $n$-vertex $r$-graph with no subhypergraph isomorphic to any Berge-$F$ is denoted $\ex_r(n,\textrm{Berge-}F)$. In this paper we establish new upper and lower bounds on $\ex_r(n,\textrm{Berge-}F)$ for general graphs $F$, and investigate connections between $\ex_r(n,\textrm{Berge-}F)$ and other recently studied extremal functions…
For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Be…
Extending a result of Rado to hypergraphs, we prove that for all $s, k, t \in \mathbb{N}$ with $k \geq t \geq 2$, the vertices of every $r = s(k-t+1)$-edge-coloured countably infinite complete $k$-graph can be partitioned into the cores of at most $s$ monochromatic $t$-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible.
The famous Dirac's Theorem gives an exact bound on the minimum degree of an $n$-vertex graph guaranteeing the existence of a hamiltonian cycle. We prove exact bounds of similar type for hamiltonian Berge cycles in $r$-uniform, $n$-vertex hypergraphs for all $3\leq r< n$. The bounds are different for $r<n/2$ and $r\geq n/2$. We also give bounds on the minimum degree guaranteeing existence of Berge cycles of length at least $k$ in such hypergraphs; the bounds are exact for all $k\geq n/2$.
Let $\mathcal{F}$ be an $r$-uniform hypergraph and $G$ be a multigraph. The hypergraph $\mathcal{F}$ is a Berge-$G$ if there is a bijection $f: E(G) \rightarrow E( \mathcal{F} )$ such that $e \subseteq f(e)$ for each $e \in E(G)$. Given a family of multigraphs $\mathcal{G}$, a hypergraph $\mathcal{H}$ is said to be $\mathcal{G}$-free if for each $G \in \mathcal{G}$, $\mathcal{H}$ does not contain a subhypergraph that is isomorphic to a Berge-$G$. We prove bounds on the maximum number of edges in an $r$-uniform linear hypergraph that is $K_{2,t}$-free. We also determine an asymptotic formula fo…
Given a graph $G$, a Berge copy of $G$ is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H ori in 2006 showed that for $r=3$ or $r=4$, an $r$-uniform $n$-vertex Berge triangle-free hypergraph has at most $\lfloor n^2/8(r-2)\rfloor$ hyperedges if $n$ is large enough, and this bound is sharp. The book graph $B_t$ consists of $t$ triangles sharing an edge. Very recently, Ghosh, Győri, Nagy-György, Paulos, Xiao and Zamora showed that a 3-uniform $n$-vertex Berge $B_t$-free hypergraph has at most $n^2/8+o(n^2)$ hyperedges if $n$ is large enough. They conjectured that this bound can be…
In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. For $r\geq 3$, we prove an optimal stopping-time result that if edges are sequently added to an initially empty $r$-graph, then as soon as the minimum degree is at least 2, the hypergraph almost surely has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős--Rényi random $r$-graph, and we also show that the $2$-out random $r$-graph almost surely has such a cycle. We obtain similar results for \textit{weak Berge} cycles as well, thus resolvi…
For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this paper, we define a variant of Turán number in hypergraphs, namely the cover Turán number, denoted as $\hat{ex}_R(n, G)$, as the maximum number of edges in the shadow graph of a Berge-$G$ free $R$-graph on $n$ vertices. We show a general…
Given a family $\mathcal{F}$ of $r$-graphs, the Turán number of $\mathcal{F}$ for a given positive integer $N$, denoted by $ex(N,\mathcal{F})$, is the maximum number of edges of an $r$-graph on $N$ vertices that does not contain any member of $\mathcal{F}$ as a subgraph. For given $r\geq 3$, a complete $r$-uniform Berge-hypergraph, denoted by { ${K}_n^{(r)}$}, is an $r$-uniform hypergraph of order $n$ with the core sequence $v_{1}, v_{2}, \ldots ,v_{n}$ as the vertices and distinct edges $e_{ij},$ $1\leq i<j\leq n,$ where every $e_{ij}$ contains both $v_{i}$ and $v_{j}$. Let $\mathcal{F}^{(r)}…
A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. Grósz, Methuku and Tompkins in 2020 showed that for any graph $F$, there is an integer $r_0=r_0(F)$, such that for any $r\ge r_0$, any $r$-uniform hypergraph without a Berge copy of $F$ has $o(n^2)$ hyperedges. The smallest such $r_0$ is called the uniformity threshold of $F$ and is denoted by $th(F)$. They showed that $th(F)\le R(F,F')$, where $R$ denotes the off-diagonal Ramsey number and $F'$ is any graph obtained form $F$ by deleting an edge. We improve this bound to $th(F)\le R(K_{χ(F)},F')$, and use the …
It has been conjectured that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every (r - 1)-coloring of the edges of the complete r-uniform hypergraph on n vertices. In this paper, we show that the statement of this conjecture is true with r-2 colors (instead of r-1 colors) by showing that there is a monochromatic Hamiltonian t-tight Berge-cycle in every b r-2 / t-1 -edge coloring of Kr n for any fixed r > t >= 2 and sufficiently large n. Also, we give a proof for this conjecture when r = 4 (the first open case). These results improve the previously…
Let $F$ be a graph. A hypergraph is called Berge $F$ if it can be obtained by replacing each edge in $F$ by a hyperedge containing it. Given a family of graphs $\mathcal{F}$, we say that a hypergraph $H$ is Berge $\mathcal{F}$-free if for every $F \in \mathcal{F}$, the hypergraph $H$ does not contain a Berge $F$ as a subhypergraph. In this paper we investigate the connections between spectral radius of the adjacency tensor and structural properties of a linear hypergraph. In particular, we obtain a spectral version of Turán-type problems over linear $k$-uniform hypergraphs by using spectral me…
In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if $n$ divides $\binom{n}{k}$, then the complete $k$-uniform hypergraph on $n$ vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an alternating sequence $v_1,e_1,v_2,\dots,v_n,e_n$ of distinct vertices $v_i$ and distinct edges $e_i$ so that each $e_i$ contains $v_i$ and $v_{i+1}$. So the divisibility condition is clearly necessary. In this note, we prove that the conjecture holds whenever $k \ge 4$ and $n \ge 30$. Our argument is based on the Kruskal-Katona theorem. The case when $k=3$ was alre…
A \emph{simple} matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a \emph{Berge hypergraph} if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $||A||$ denote the number of columns in $A$, we define the extremal function $Bh(m,{ F})=\max\{||A||\,:\, A \hbox{ is }m\hbox{-rowed simple matrix with no Berge hypergraph }F\}$. We determine the asymptotics of $Bh(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determini…
For a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $G$ (or a Berge-$G$ in short), if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We denote the family of $r$-uniform hypergraphs that are Berge copies of $G$ by $B^rG$. For families of $r$-uniform hypergraphs $\mathbf{H}$ and $\mathbf{H}'$, we denote by $R(\mathbf{H},\mathbf{H}')$ the smallest number $n$ such that in any blue-red coloring of $\mathcal{K}_n^r$ (the complete $r$-uniform hypergraph on $n$ vertices) there is a monochromatic blue copy of a hypergrap…
In this paper we study the maximum number of hyperedges which may be in an $r$-uniform hypergraph under the restriction that no pair of vertices has more than $t$ Berge paths of length $k$ between them. When $r=t=2$, this is the even-cycle problem asking for $\mathrm{ex}(n, C_{2k})$. We extend results of Füredi and Simonovits and of Conlon, who studied the problem when $r=2$. In particular, we show that for fixed $k$ and $r$, there is a constant $t$ such that the maximum number of edges can be determined in order of magnitude.
In 1981, Alspach conjectured that the complete graph $ K_{n} $ could be decomposed into cycles of arbitrary lengths, provided that the obvious necessary conditions would hold. This conjecture was proved completely by Bryant, Horsley and Pettersson in 2014. Moreover, in 1983, Tarsi conjectured that the obvious necessary conditions for packing pairwise edge-disjoint paths of arbitrary lengths in the complete multigraphs were also sufficient. The conjecture was confirmed by Bryant in 2010. In this paper, we investigate an analogous problem as the decomposition of the complete uniform multi-hyperg…
For an arbitrary graph $G$, a hypergraph $\mathcal{H}$ is called Berge-$G$ if there is a bijection $Φ:E(G)\longrightarrow E( \mathcal{H})$ such that for each $e\in E(G)$, we have $e\subseteq Φ(e)$. We denote by $\mathcal{B}^rG$, the family of $r$-uniform Berge-$G$ hypergraphs. For families $\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t$ of $r$-uniform hypergraphs, the Ramsey number $R(\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t)$ is the smallest integer $n$ such that in every $t$-hyperedge coloring of $\mathcal{K}_{n}^r$ there is a monochromatic copy of a hypergraph in $\mathcal{H}…
A classical conjecture of Erdős and Sós asks to determine the Turán number of a tree. We consider variants of this problem in the settings of hypergraphs and multi-hypergraphs. In particular, for all $k$ and $r$, with $r \ge k (k-2)$, we show that any $r$-uniform hypergraph $\mathcal{H}$ with more than $\frac{n(k-1)}{r+1}$ hyperedges contains a Berge copy of any tree with $k$ edges different from the $k$-edge star. This bound is sharp when $r+1$ divides $n$ and for such values of $n$ we determine the extremal hypergraphs.
A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$ and $\{v_i, v_{i+1}\} \subseteq e_i$ for every $i\in [n-1]$. We prove the following Dirac-type theorem about Berge cycles in the binomial random $r$-uniform hypergraph $H^{(r)}(n,p)$: for every integer $r \geq 3$, every real $γ>0$ and $p \geq \frac{\ln^{17r} n}{n^{r-1}}$ asymptotically almost surely, every spanning subgraph $H \subseteq H^{(r)}(n,p)$ with mini…
Let $F = (U,E)$ be a graph and $\mathcal{H} = (V,\mathcal{E})$ be a hypergraph. We say that $\mathcal{H}$ contains a Berge-$F$ if there exist injections $ψ:U\to V$ and $φ:E\to \mathcal{E}$ such that for every $e=\{u,v\}\in E$, $\{ψ(u),ψ(v)\}\subsetφ(e)$. Let $ex_r(n,F)$ denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$ vertices which does not contain a Berge-$F$. For small enough $r$ and non-bipartite $F$, $ex_r(n,F)=Ω(n^2)$; we show that for sufficiently large $r$, $ex_r(n,F)=o(n^2)$. Let $thres(F) = \min\{r_0 :ex_r(n,F) = o(n^2) \text{ for all } r \ge r_0 \}$. We sh…
Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure of extremal hypergraphs for the case when $k-2$ divides $n-1$ and $k\geq r+3$. In this paper we determine $EG_r(n,k)$ and describe the extremal hypergraphs for all $n$ when $k\geq r+4$.
Let $n\geq k\geq r+3$ and $\mathcal H$ be an $n$-vertex $r$-uniform hypergraph. We show that if $|\mathcal H|> \frac{n-1}{k-2}\binom{k-1}{r}$ then $\mathcal H$ contains a Berge cycle of length at least $k$. This bound is tight when $k-2$ divides $n-1$. We also show that the bound is attained only for connected $r$-uniform hypergraphs in which every block is the complete hypergraph $K^{(r)}_{k-1}$. We conjecture that our bound also holds in the case $k=r+2$, but the case of short cycles, $k\leq r+1$, is different.
Berge Hamilton cycles in a random sparsification of dense hypergraphs
Center for AI and Natural Sciences, Korea Institute for Advanced Study (KIAS)（韩国高级科学研究院人工智能与自然科学中心）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极端组合与概率研究组）；Department of Mathematical Sciences, KAIST（韩国科学技术院数学系）
For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. An $R$-graph $\mathcal{H}$ is \emph{covering} if every vertex pair of $\mathcal{H}$ is contained in some hyperedge. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this note, we define a new type of Ramsey number, namely the \emph{cover Ramsey number}, denoted as $\hat{R}^R(BG_1…
In this paper, we develop a method for studying cycle lengths in hypergraphs. Our method is built on earlier ones used in [21,22,18]. However, instead of utilizing the well-known lemma of Bondy and Simonovits [4] that most existing methods do, we develop a new and very simple lemma in its place. One useful feature of the new lemma is its adaptiveness for the hypergraph setting. Using this new method, we prove a conjecture of Verstraëte [37] that for $r\ge 3$, every $r$-uniform hypergraph with average degree $Ω(k^{r-1})$ contains Berge cycles of $k$ consecutive lengths. This is sharp up to the …
