We describe a method that we believe may be foundational for a comprehensive theory of generalised Turan problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2-q+1 and n/2, but for PG_2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2-c)n for some small c>0. We conclude by setting out a program for future developments of this method to tackle other problems.
In this paper, we consider the Turán problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\}, \{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Turán densities of some small $\{1,3\}$-hypergraphs.
Given a family of $k$-hypergraphs $\mathcal{F}$, $ex(n,\mathcal{F})$ is the maximum number of edges a $k$-hypergraph can have, knowing that said hypergraph has $n$ vertices but contains no copy of any hypergraph from $\mathcal{F}$ as a subgraph. We prove that for every rational $r$ between $0$ and $k-1$, there exists some finite family $\mathcal{F}$ of $k$-hypergraphs for which $ex(n,\mathcal{F})=Θ(n^{k-r})$.
A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the tetrahedron exhibits this phenomenon assuming Turán's conjecture. Our main result is to construct a finite family of triple systems $\mathcal{M}$, determine its Turán number, and prove that there are two near-extremal $\mathcal{M}$-free constructions that are far from each other in edit-distance. This is the first extremal result for a hypergraph family that …
A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Turán density $π(H)$ is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying $R(G_n)\subseteq R(H)$, and $h_n(G_n)$, the so called Lubell function, is the expected number of edges in $G_n$ hit by a random full chain. This …
Let l>k>2. Let the k-graph H^k_l be obtained from the complete 2-graph K_l^2 by enlarging each edge with a new set of k-2 vertices. Mubayi ["A hypergraph extension of Turan's theorem", to appear in J. Combin. Th. (B)] computed asymptotically the Turan function ex(n,H_l^k). Here we determine the exact value of ex(n,H_l^k) for all sufficiently large n, settling a conjecture of Mubayi.
Recently, several hypergraph Turán problems were solved by the powerful random algebraic method. However, the random algebraic method usually requires some parameters to be very large, hence we are concerned about how these Turán numbers depend on such large parameters of the forbidden hypergraphs. In this paper, we determine the dependence on such specified large constant for several hypergraph Turán problems. More specifically, for complete $r$-partite $r$-uniform hypergraphs, we show that if $s_{r}$ is sufficiently larger than $s_{1},s_{2},\ldots,s_{r-1},$ then $$\textup{ex}_{r}(n,K_{s_{1},…
Fix a graph $F$. We say that a graph is {\it $F$-free} if it does not contain $F$ as a subgraph. The {\it Turán number} of $F$, denoted $\mathrm{ex}(n,F)$, is the maximum number of edges possible in an $n$-vertex $F$-free graph. The study of Turán numbers is a central problem in graph theory. The goal of this paper is to generalize a theorem of Lidický, Liu and Palmer [{\it Electron.\ J.\ of Combin.}\ {\bf 20} (2016)] that determines $\mathrm{ex}(n,F)$ for $F$ a forest of stars. In particular, we consider generalizations of the problem to three different well-studied hypergraph settings and in…
In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least $d$ contains $H$. In particular, they raise the questions of determining the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. The former question was solved only recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter sti…
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)}…
We consider the Turán problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$ and $E_b$ do not have to be disjoint. The Turán density $π(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/{n\choose 2}$ is the formula to measure the edge density of $G_n$. We will determine the Turán densities of…
Let $F$ be a graph. We say that a hypergraph $H$ contains an induced Berge $F$ if the vertices of $F$ can be embedded to $H$ (e.g., $V(F)\subseteq V(H)$) and there exists an injective mapping $f$ from the edges of $F$ to the hyperedges of $H$ such that $f(xy) \cap V(F) = \{x,y\}$ holds for each edge $xy$ of $F$. In other words, $H$ contains $F$ as a trace. Let $ex_{r}(n,B_{ind} F)$ denote the maximum number of edges in an $r$-uniform hypergraph with no induced Berge $F$. Let $ex(n,K_r, F)$ denote the maximum number of $K_r$'s in an $F$-free graph on $n$ vertices. We show that these two Turán t…
This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Seco…
Let $G$ be a cancellative $3$-uniform hypergraph in which the symmetric difference of any two edges is not contained in a third one. Equivalently, a $3$-uniform hypergraph $G$ is cancellative if and only if $G$ is $\{F_4, F_5\}$-free, where $F_4 = \{abc, abd, bcd\}$ and $F_5 = \{abc, abd, cde\}$. A classical result in extremal combinatorics stated that the maximum size of a cancellative hypergraph is achieved by the balanced complete tripartite $3$-uniform hypergraph, which was firstly proved by Bollobás and later by Keevash and Mubayi. In this paper, we consider spectral extremal problems for…
For integers $q\ge p\ge r\ge2$, we say that an $r$-uniform hypergraph $H$ has property $(q,p)$, if for any $q$-vertex subset $Q$ of $V(H)$, there exists a $p$-vertex subset $P$ of $Q$ spanning a clique in $H$. Let $T_{r}(n,q,p)=\min\{ e(H): H\subset \binom{[n]}{r}, H \text{~has property~} (q,p)\}$. The local Turán density about property $(q,p)$ in $r$-uniform hypergraphs is defined as $t_{r}(q,p)=\lim_{n\to \infty}T_{r}(n,q,p)/\binom{n}{r}$. Frankl, Huang and Rödl [J. Comb. Theory, Ser. A, 177 (2021)] showed that $\lim_{p\to\infty}t_{r}(ap+1,p+1)=\frac{1}{a^{r-1}}$ for positive integer $a$ and…
The $r$-expansion $G^+$ of a graph $G$ is the $r$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a vertex subset of size $r-2$ disjoint from $V(G)$ such that distinct edges are enlarged by disjoint subsets. Let $ex_r(n,F)$ denote the maximum number of edges in an $r$-uniform hypergraph with $n$ vertices not containing any copy of the $r$-uniform hypergraph $F$. Many problems in extremal set theory ask for the determination of $ex_r(n,G^+)$ for various graphs $G$. We survey these Turán-type problems, focusing on recent developments.
Given a positive integer $n$ and an $r$-uniform hypergraph (or $r$-graph for short) $F$, the Turan number $ex(n,F)$ of $F$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain $F$ as a subgraph. The extension $H^F $ of $F$ is obtained as follows: For each pair of vertices $v_i,v_j$ in $F$ not contained in an edge of $F$, we add a set $B_{ij}$ of $r-2$ new vertices and the edge $\{v_i,v_j\} \cup B_{ij}$, where the $B_{ij}$ 's are pairwise disjoint over all such pairs $\{i,j\}$. Let $K^r_p$ denote the complete $r$-graph on $p$ vertices. For all sufficiently large …
Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron $K_4^3$. This is equivalent to determining the maximum $\ell_1$-norm of the codegree vector of a $K_4^3$-free $n$-vertex $3$-uniform hypergraph. We introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum, $\text{co}_2(G)$, of a $3$-uniform hypergraph $G$ is the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $xy$, or in other words, the square of the $\ell_2$-norm of the codegree v…
Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting $P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all sets $P_i \cup P_j$ with $i \neq j$. This can be thought of as the `$k$-expansion' of the complete graph $K_r$: each vertex has been replaced with a set of size $k$. We determine the exact Turan number of $C^{2k}_3$ and the corresponding extremal hypergraph, thus confirming a conjecture of Frankl. Sidorenko has given an upper bound of $(r-2) / (r-1)$ for the Turán density of $C^{2k}_r$ for any $r$, and a construction establishing a matching lower bound w…
Confirming a conjecture of Vera T. Sós in a very strong sense, we give a complete solution to Turán's hypergraph problem for the Fano plane. That is we prove for $n\ge 8$ that among all $3$-uniform hypergraphs on $n$ vertices not containing the Fano plane there is indeed exactly one whose number of edges is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for $n=7$ there is exactly one other extremal configuration with the same number of edges: the hypergraph arising from a clique of order $7$ by removing all five edges containing a fixed pair of vertices. For sufficient…
For two graphs $F$ and $H$, the relative Turán number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these quantities as a function of the maximum degree of $H$. In this paper, we study a generalization for uniform hypergraphs. If $F$ is a complete $r$-partite $r$-uniform hypergraph with parts of sizes $s_1,s_2,\dots,s_r$ with each $s_{i + 1}$ sufficiently large relative to $s_i$, then with $1/β= \sum_{i = 2}^r \prod_{j = 1}^{i - 1} s_j$ we prove that for any $r$-uniform hyp…
Given a $k$-graph $H$ a complete blow-up of $H$ is a $k$-graph $\hat{H}$ formed by replacing each $v\in V(H)$ by a non-empty vertex class $A_v$ and then inserting all edges between any $k$ vertex classes corresponding to an edge of $H$. Given a subgraph $G\subseteq \hat{H}$ and an edge $e\in E(H)$ we define the density $d_e(G)$ to be the proportion of edges present in $G$ between the classes corresponding to $e$. The density Turán problem for $H$ asks: determine the minimal value $d_{crit}(H)$ such that any subgraph $G\subseteq \hat{H}$ satisfying $d_e(G)> d_{crit}(H)$ for every $e\in E(H)$ co…
A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extensions were applied in Turán problems of graphs and uniform hypergraphs. Very recently, the study of Turán densities of non-uniform hypergraphs has been motivated by extremal poset problems. In this paper, we give some Motzkin-Straus type results for non-uniform hypergraphs.
Write $K^{(k)}_{n}$ for the complete $k$-graph on $n$ vertices. For $2 \leq k \leq g < r$ integers, let $π\left(n, K^{(k)}_{g}, K^{(k)}_r\right)$ be the maximum density of $K^{(k)}_{g}$ in $n$ vertex $K^{(k)}_{r}$-free $k$-graphs. The main contribution of this paper is the upper bound: $π\left(n, K^{(k)}_{g}, K^{(k)}_r\right) \leq \left(1 + O\left(n^{-1}\right) \right)\prod_{m=k}^{g} \left(1 - \frac{\binom{m-1}{k-1}}{\binom{r-1}{k-1}} \right).$ The graph case ($k=2$) is the first known generalized Turán question, investigated by Erdős. The $k=g$ case is the hypergraph Turán problem where the b…
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…
A $k$-uniform linear path of length $\ell$, denoted by $P^{(k)}_\ell$, is a family of $k$-sets $\{F_1,..., F_\ell\}$ such that $|F_i\cap F_{i+1}|=1$ for each $i$ and $F_i\cap F_j=\emptyset$ whenever $|i-j|>1$. Given a $k$-uniform hypergraph $H$ and a positive integer $n$, the {\it $k$-uniform hypergraph Turán number} of $H$, denoted by $\ex_k(n,H)$, is the maximum number of edges in a $k$-uniform hypergraph $\cF$ on $n$ vertices that does not contain $H$ as a subhypergraph. With an intensive use of the delta-system method, we determine $\ex_k(n,P^{(k)}_\ell)$ exactly for all fixed $\ell\geq 1,…
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…
Since its formulation, Turán's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a $3$-uniform hypergraph $\mathcal{F}$ on $n$ vertices in which any five vertices span at least one edge, prove that $|\mathcal{F}| \ge (1/4 -o(1))\binom{n}{3}$. The construction showing that this bound would be best possible is simply $\binom{X}{3} \cup \binom{Y}{3}$ where $X$ and $Y$ evenly partition the vertex set. This construction has the following more general $(2p+1, p+1)$-property: any set of $2p+1$ vertices spans a complet…
Upper Bounds on Turán Densities via Extremal Set Theory
Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
An $r$-uniform hypergraph is called $t$-cancellative if for any $t+2$ distinct edges $A_1,\ldots,A_t,B,C$, it holds that $(\cup_{i=1}^t A_i)\cup B\neq (\cup_{i=1}^t A_i)\cup C$. It is called $t$-union-free if for any two distinct subsets $\mathcal{A},\mathcal{B}$, each consisting of at most $t$ edges, it holds that $\cup_{A\in\mathcal{A}} A\neq \cup_{B\in\mathcal{B}} B$. Let $C_t(n,r)$ (resp. $U_t(n,r)$) denote the maximum number of edges of a $t$-cancellative (resp. $t$-union-free) $r$-uniform hypergraph on $n$ vertices. Among other results, we show that for fixed $r\ge 3,t\ge 3$ and $n\right…
Vertex-colored Turán theorems with applications in extremal hypergraph problems
School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；School of Data Science and Engineering, South China Normal University（华南师范大学数据科学与工程学院）；Center for Discrete Mathematics, Fuzhou University（福州大学离散数学研究中心）
Extremal problems for $3$-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is weakly $(d,η)$-quasirandom if for any subset $U\subseteq V$ the number of hyperedges of $H$ contained in $U$ is in the interval $d\binom{|U|}{3}\pmη|V|^3$. We show that for any $\varepsilon>0$ there exists $η>0$ such that every sufficiently large weakly $(1/4+\varepsilon,η)$-quasirandom hypergraph contains four v…
We make some conjectures about extremal densities of daisy-free families, where a `daisy' is a certain hypergraph. These questions turn out to be related to some Turan problems in the hypercube, but they are also natural in their own right.
For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of $\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. The determination of Turán number $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ has become the fundamental core problem in extremal graph theory ever since the pioneering work Turán's Theorem was published in $1941$. Although we have some rich results for the simple graph case, only sporadic results have been known for the hypergraph Turán problems. In this paper, we mainly focus on the function $\text{ex…
In this paper we define a class of combinatorial structures the instances of which can each be thought of as a model of directed hypergraphs in some way. Each of these models is uniform in that all edges have the same internal structure, and each is simple in that no loops or multiedges are allowed. We generalize the concepts of Turan density, blowup density, and jumps to this class and show that many basic extremal results extend naturally in this new setting. In particular, we show that supersaturation holds, the blowup of a generalized directed hypergraph (GDH) has the same Turan density as…
For positive integers $s,t,r$, let $K_{s,t}^{(r)}$ denote the $r$-uniform hypergraph whose vertex set is the union of pairwise disjoint sets $X,Y_1,\dots,Y_t$, where $|X| = s$ and $|Y_1| = \dots = |Y_t| = r-1$, and whose edge set is $\{\{x\} \cup Y_i: x \in X, 1\leq i\leq t\}$. The study of the Turán function of $K_{s,t}^{(r)}$ received considerable interest in recent years. Our main results are as follows. First, we show that \begin{equation} \mathrm{ex}(n,K_{s,t}^{(r)}) = O_{s,r}(t^{\frac{1}{s-1}}n^{r - \frac{1}{s-1}}) \end{equation} for all $s,t\geq 2$ and $r\geq 3$, improving the power of …
A k-uniform linear cycle of length s is a cyclic list of k-sets A_1,..., A_s such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k at least 5 and s at least 3 and sufficiently large n we determine the largest size of a k-uniform set family on [n] not containing a linear cycle of length s. For odd s=2t+1 the unique extremal family F_S consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even s=2t+2, the unique extremal family consists of F_S plus all the k-sets outside S containing some fixed two elements. For k at least 4 …
For fixed integers $r\ge 3, e\ge 3$, and $v\ge r+1$, let $f_r(n,v,e)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph in which the union of arbitrary $e$ distinct edges contains at least $v+1$ vertices. In 1973, Brown, Erdős and Sós proved that $f_r(n,er-(e-1)k,e)=Θ(n^k)$ and conjectured that the limit $\lim_{n\rightarrow\infty}\frac{f_3(n,e+2,e)}{n^2}$ always exists for all fixed integers $e\ge 3$. In 2020 Shangguan and Tamo conjectured that the limit $\lim_{n\rightarrow\infty}\frac{f_r(n,er-(e-1)k,e)}{n^k}$ always exists for all fixed integers $r>k\ge 2$ and $e\ge …
