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A tight $\ell$-cycle minus an edge $C_\ell^-$ is the $3$-graph on the vertex set $[\ell]$, where any three consecutive vertices in the string $123\ldots\ell 1$ form an edge. We show that for every $\ell\ge 5$, $\ell$ not divisible by $3$, the extremal number is $ ex\left(C_\ell^-,n\right)=\tfrac1{24}n^3+O(n\ln n)=\left(\tfrac14+o(1)\right){n\choose 3}. $ We determine the extremal graph up to $O(n)$ edge edits.

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Tur\'{a}n density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\alpha$ such that there exist arbitrarily large $n$-vertex $F$-free $k$-graphs $H$ in which every $(k-1)$-subset of $V(H)$ is contained in at least $\alpha n$ edges. Recently, it was proved that for every $3$-graph $F$, $\pi_{\mathrm{co}}(F)=0$ implies $\pi_{\therefore}(F)=0$, where $\pi_{\therefore}(F)$ is the uniform Tur\'{a}n density of $F$ and is defined as the supremum over all $d$ such that there are infinitely many $F$-free $k$-graphs $H$ satisfying that any induced linear-size subhypergraph of $H$ has edge density at least $d$. In this paper, we introduce a layered structure for $3$-graphs which allows us to obtain the reverse implication: every layered $3$-graph $F$ with $\pi_{\therefore}(F)=0$ satisfies $\pi_{\mathrm{co}}(F)=0$. Along the way, we answer in the negative a question of Falgas-Ravry, Pikhurko, Vaughan and Volec [J. London Math. Soc., 2023] about whether $\pi_{\therefore}(F)\leq\pi_{\mathrm{co}}(F)$ always holds. In particular, we construct counterexamples $F$ with positive but arbitrarily small $\pi_{\mathrm{co}}(F)$ while having $\pi_{\therefore}(F)\ge 4/27$., Comment: 17 pages. This work will be merged with arXiv:2312.02879

Let $F$ be a graph and let $\mathcal{B}_r(F)$ be the class of $r$-uniform Berge-$F$ hypergraphs. In this paper, by establishing a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure via walks, we give a spectral asymptotic bound for $\mathcal{B}_{r}(C_3)$-free linear $r$-uniform hypergraphs and upper bounds for the spectral radii of $\mathcal{B}_{r}(K_{2,t})$-free or $\{\mathcal{B}_{r}(K_{s,t}),\mathcal{B}_{r}(C_{3})\}$-free linear $r$-uniform hypergraphs, where $C_{3}$ and $K_{s,t}$ are respectively the triangle and the complete bipartite graph with one part having $s$ vertices and the other part having $t$ vertices. Our work implies an upper bound for the number of edges of $\{\mathcal{B}_{r}(K_{s,t}),\mathcal{B}_{r}(C_{3})\}$-free linear $r$-uniform hypergraphs, and extends some known work on (spectral) extreme problems of hypergraphs.

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Tur\'{a}n density $\pi_{\mathrm{co}}(F)$ is the infimum over all $\alpha$ such that any $n$-vertex $k$-graph $H$ with every $(k-1)$-subset of $V(H)$ contained in at least $\alpha n$ edges has a copy of $F$. The uniform Tur\'{a}n density $\pi_{\therefore}(F)$ is the supremum over all $d$ such that there are infinitely many $F$-free $k$-graphs $H$ satisfying that any linear-size subhypergraph of $H$ has edge density at least $d$. Falgas-Ravry, Pikhurko, Vaughan and Volec [J. London Math. Soc., 2023] asked whether for every $3$-graph $F$, $\pi_{\therefore}(F)\leq\pi_{\mathrm{co}}(F)$. We provide a positive answer to this question provided that $\pi_{\mathrm{co}}(F)=0$. Our proof relies on a random geometric construction and a new formulation of the characterization of $3$-graphs with vanishing uniform Tur\'{a}n density due to Reiher, R{\"o}dl and Schacht [J. London Math. Soc., 2018]. Along the way, we answer a question of Falgas-Ravry, Pikhurko, Vaughan and Volec about subhypergraphs with linear minimum codegree in uniformly dense hypergraphs in the negative., Comment: 10 pages

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniform hypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subset of size $r-2$ disjoint from the vertex set of $F$ such that distinct edges are enlarged by disjoint subsets. Let $ex_{r}^{lin}(n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ be the maximum number of edges and the maximum spectral radius of all $F^{r}$-free linear $r$-uniform hypergraphs with $n$ vertices, respectively. In this paper, we present the sharp (or asymptotic) bounds of $ex_{r}^{lin}( n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ by establishing the connection between the spectral radii of linear hypergraphs and those of their shadow graphs, where $F$ is a $(k+1)$-color critical graph or a graph with chromatic number $k$.

Let $H_k^r$ denote an $r$-uniform hypergraph with $k$ edges and $r+1$ vertices, where $k \leq r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Tur\'{a}n density are $\pi(H_k^r) \leq \frac{k-2}{r}$ for all $k \geq 3$, and $\pi(H_3^r) \geq 2^{1-r}$ for $k=3$. We prove that $\pi(H_k^r) \geq (C_k - o(1)) \, r^{-(1+\frac{1}{k-2})}$ as $r\to\infty$. In the case $k=3$, we prove $\pi(H_3^r) \geq (1.7215 - o(1)) \, r^{-2}$ as $r\to\infty$, and $\pi(H_3^r) \geq r^{-2}$ for all $r$.

The Tur\'{a}n problem asks for the largest number of edges ex$(n,H)$ in an $n$-vertex graph not containing a fixed forbidden subgraph $H$, which is one of the most important problems in extremal graph theory. However the order of magnitude of ex$(n,H)$ for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomail resultant approach to algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lowers bounds for Tur\'{a}n number of $1$-subdivision of $K_{3,t_{1}}$ and linear Tur\'{a}n number of Berge theta hyerpgraph $\Theta_{3,t_{2}}^{B}$ with $t_{1}=25$ and $t_{2}=217$. Moreover, the constant $t_{1}$ improves the random algebraic construction of Bukh and Conlon~[Rational exponents in extremal graph theory, J. Eur. Math. Soc. 20 (2018), 1747-1757] and makes progress on the known estimation for the smallest value of $t_{1}$ concerning a problem posed by Conlon, Janzer and Lee ~[More on the extremal number of subdivisions, Combinatorica, to appear], while the constant $t_{2}$ improves a result of He and Tait~[Hypergraphs with few berge paths of fixed length between vertices, SIAM J. Discrete Math., 33(3), 1472-1481]., Comment: 46 pages. arXiv admin note: substantial text overlap with arXiv:2002.06795, arXiv:1908.01459

We consider the Tur\'an 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 Tur\'an density $\pi(H)$ 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 Tur\'an densities of all $2$-edge-colored bipartite graphs. We also give an important application of our study on the Tur\'an problems of $\{2, 3\}$-hypergraphs., Comment: 21 pages

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The Tur\'{a}n number $\text{ex}(n,\mathcal{H})$ is the maximum number of edges in an $n$-vertex $\mathcal{H}$-free $r$-uniform hypergraph. The Tur\'{a}n density of $\mathcal{H}$ is defined by \[\pi(\mathcal{H})=\lim_{n\rightarrow\infty}\frac{\text{ex}(n,\mathcal{H})}{\binom{n}{r}}.\] In this paper, we consider the Tur\'{a}n density of projective geometries. We give two new constructions of $PG_{m}(q)$-free hypergraphs which improve some results given by Keevash (J. Combin. Theory Ser. A, 111: 289--309, 2005). Based on an upper bound of blocking sets of $PG_m(q)$, we give a new general lower bound for the Tur\'{a}n density of $PG_{m}(q)$. By a detailed analysis of the structures of complete arcs in $PG_2(q)$, we also get better lower bounds for the Tur\'{a}n density of $PG_2(q)$ with $q=3,\ 4,\ 5,\ 7,\ 8$., Comment: 15 pages

Given two $3$-graphs $F$ and $H$, an $F$-covering of $H$ is a collection of copies of $F$ in $H$ such that each vertex of $H$ is contained in at least one copy of them. Let {$c_2(n,F)$} be the maximum integer $t$ such that every 3-graph with minimum codegree greater than $t$ has an $F$-covering. In this note, we answer an open problem of Falgas-Ravry and Zhao (SIAM J. Discrete Math., 2016) by determining the exact value of {$c_2(n, K_4^-)$} and {$c_2(n, K_5^-)$}, where $K_t^-$ is the complete $3$-graph on $t$ vertices with one edge removed., Comment: 9 pages

Recently, Berge theta hypergraphs have received special attention due to the similarity with Berge even cycles. Let $r$-uniform Berge theta hypergraph $\Theta_{\ell,t}^{B}$ 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 Tur\'{a}n 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,\Theta_{3,217}^{B})=\Omega(n^{\frac{4}{3}}). \end{align*} This matches an earlier upper bound by He and Tait up to an absolute constant factor. The construction is algebraic, which is based on some equations over finite fields, and the parameter $t$ in our construction is much smaller than that in random algebraic construction. Our main technique is using the resultant of polynomials, which appears to be a powerful technique to eliminate variables., Comment: 19 pages

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 Tur\'{a}n number $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ has become the fundamental core problem in extremal graph theory ever since the pioneering work Tur\'{a}n'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 Tur\'{a}n problems. In this paper, we mainly focus on the function $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ when $\mathcal{H}$ is one of two different hypergraph extensions of the complete bipartite graph $K_{s,t}$. The first extension is the complete bipartite $r$-graph $K_{s,t}^{(r)}$, which was introduced by Mubayi and Verstra\"{e}te~[J. Combin. Theory Ser. A, 106: 237--253, 2004]. Using the powerful random algebraic method, we show that if $s$ is sufficiently larger than $t$, then \[\text{ex}_{r}(n,\mathcal{T},K_{s,t}^{(r)})=\Omega(n^{v-\frac{e}{t}}),\] where $\mathcal{T}$ is an $r$-graph with $v$ vertices and $e$ edges. In particular, when $\mathcal{T}$ is an edge or some specified complete bipartite $r$-graph, we can determine their asymptotics. The second important extension is the complete $r$-partite $r$-graph $K_{s_{1},s_{2},\ldots,s_{r}}^{(r)}$, which has been widely studied. When $r=3$, we provide an explicit construction giving \[\text{ex}_{3}(n,K_{2,2,7}^{(3)})\geqslant\frac{1}{27}n^{\frac{19}{7}}+o(n^{\frac{19}{7}}).\] Our construction is based on the Norm graph, and improves the lower bound $\Omega(n^{\frac{73}{27}})$ obtained by probabilistic method., Comment: To appear in SCIENCE CHINA Mathematics

A result due to Gy\'arf\'as, Hubenko, and Solymosi (answering a question of Erd\"os) states that if a graph $G$ on $n$ vertices does not contain $K_{2,2}$ as an induced subgraph yet has at least $c\binom{n}{2}$ edges, then $G$ has a complete subgraph on at least $\frac{c^2}{10}n$ vertices. In this paper we suggest a "higher-dimensional" analogue of the notion of an induced $K_{2,2}$ which allows us to generalize their result to $k$-uniform hypergraphs. Our result also has an interesting consequence in discrete geometry. In particular, it implies that the fractional Helly theorem can be derived as a purely combinatorial consequence of the colorful Helly theorem.

For a fixed positive integer $n$ and an $r$-uniform hypergraph $H$, the Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices, and the Lagrangian density of $H$ is defined as $\pi_{\lambda}(H)=\sup \{r! \lambda(G) : G \;\text{is an}\; H\text{-free} \;r\text{-uniform hypergraph}\}$, where $\lambda(G)$ is the Lagrangian of $G$. For an $r$-uniform hypergraph $H$ on $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$. We say that an $r$-uniform hypergraph $H$ on $t$ vertices is perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$. Let $P_t=\{e_1, e_2, \dots, e_t\}$ be the linear $3$-uniform path of length $t$, that is, $|e_i|=3$, $|e_i \cap e_{i+1}|=1$ and $e_i \cap e_j=\emptyset$ if $|i-j|\ge 2$. We show that $P_3$ and $P_4$ are perfect, this supports a conjecture in \cite{yanpeng} proposing that all $3$-uniform linear hypergraphs are perfect. Applying the results on Lagrangian densities, we determine the Tur\'an numbers of their extensions., Comment: 17 pages, 6 figures. arXiv admin note: text overlap with arXiv:1609.08983; text overlap with arXiv:1510.03461 by other authors

A tight $r$-tree $T$ is an $r$-uniform hypergraph that has an edge-ordering $e_1, e_2, \dots, e_t$ such that for each $i\geq 2$, $e_i$ has a vertex $v_i$ that does not belong to any previous edge and $e_i-v_i$ is contained in $e_j$ for some $j

For a hypergraph $H$, let $q(H)$ denote the expected number of monochromatic edges when the color of each vertex in $H$ is sampled uniformly at random from the set of size 2. Let $s_{\min}(H)$ denote the minimum size of an edge in $H$. Erd\H{o}s asked in 1963 whether there exists an unbounded function $g(k)$ such that any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq g(k)$ is two colorable. Beck in 1978 answered this question in the affirmative for a function $g(k) = \Theta(\log^* k)$. We improve this result by showing that, for an absolute constant $\delta>0$, a version of random greedy coloring procedure is likely to find a proper two coloring for any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq \delta \cdot \log k$.

In this paper, we consider the Tur\'an 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 Tur\'an densities of some small $\{1,3\}$-hypergraphs., Comment: 18 pages

Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let $M_t^r$ be the $r$-graph with $t$ pairwise disjoint edges. Given an $r$-graph $F$ and a positive integer $p\geq |V(F)|$, we define the {\em extension} of $F$, denoted by $H_{p}^{F}$ as follows: Label the vertices of $F$ as $v_1,\dots,v_{|V(F)|}$. Add new vertices $v_{|V(F)|+1},\dots,v_{p}$. For each pair of vertices $v_i,v_j, 1\le i

In this paper extremal problems for uniform hypergraphs are studied in the general setting of hereditary properties. It turns out that extremal problems about edges are particular cases of a general analyic problem about a recently introduced graph parameter. The paper builds a basis for the systematic study of this parameter and illustrates a range of various proof tools. It is shown that extremal problems about the number of edges of uniform hypergraphs are asymptotically equivalent to extremal problems about the largest eigenvalue; this result is new even for 2-graphs. Several concrete problems are adressed and solutions to many more are suggested. A number of open problems are raised and directions for further studies are outlined., Comment: 31 pages, the main Theorem 12 is extended, the presentation is simplified

In this paper, we prove several new Tur\'an density results for 3-graphs with independent neighbourhoods. We show: \pi(K_4^-, C_5, F_{3,2})=12/49, \pi(K_4^-, F_{3,2})=5/18, and \pi(J_4, F_{3,2})=\pi(J_5, F_{3,2})=3/8, where J_t is the 3-graph consisting of a single vertex x together with a disjoint set A of size t and all $\binom{|A|}{2}$ 3-edges containing x. We also prove two Tur\'an density results where we forbid certain induced subgraphs: \pi(F_{3,2}, induced K_4^-)=3/8 and \pi(K_5, 5-set spanning 8 edges)=3/4. The latter result is an analogue for K_5 of Razborov's result that \pi(K_4, 4-set spanning 1 edge)=5/9. We give several new constructions, conjectures and bounds for Tur\'an densities of 3-graphs which should be of interest to researchers in the area. Our main tool is `Flagmatic', an implementation of Razborov's flag algebra calculus, which we are making publicly available. In a bid to make the power of Razborov's method more widely accessible, we have tried to make Flagmatic as user-friendly as possible, hoping to remove thereby the major hurdle that needs to be cleared before using the flag algebra calculus. Finally, we spend some time reflecting on the limitations of our approach, and in particular on which problems we may be unable to solve. Our discussion of the `complexity barrier' for the flag algebra calculus may be of general interest., Comment: 31 pages, 5 figures; version 2 corrects some minor mistakes

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., Comment: 43 pages, 4 figures