We show that for every $k \in \mathbb{N}$ there exists $C > 0$ such that if $p^k \ge C \log^8 n / n$ then asymptotically almost surely the random graph $G_{n,p}$ contains the $k$\textsuperscript{th} power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi-polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi-polynomial algorithm for finding a tight Hamilton cycle in the random $k$-uniform hypergraph $G_{n,p}^{(k)}$ for $p \ge C \log^8 n/ n$.
The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of $p$. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.
We prove that any $3$-uniform hypergraph whose minimum vertex degree is at least $\left(\frac{5}{9} + o(1) \right)\binom{n}{2}$ admits an almost-spanning tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, Böttcher, Cooley and Mycroft.
In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz [12], Sutherland [19], Kupitz and Perles [16] for convex geometric graphs, as well as the classical Erdős-Gallai Theorem [6] for graphs. As a consequence, we obtain the first substantial improvement on the Turán problem for tight paths in uniform hypergraphs.
We show that for every η > 0 there exists an integer n_0 such that every 2-colouring of the 3-uniform complete hypergraph on n \geq n_0 vertices contains two disjoint monochromatic tight cycles of distinct colours that together cover all but at most ηn vertices. The same result holds if we replace tight cycles with loose cycles.
In this paper we show that $e/n$ is the sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs, for all $k\ge 4$. When $k=3$ we show that $1/n$ is an asymptotic threshold. We also determine thresholds for the existence of other types of Hamilton cycles.
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on…
We show that every 4-uniform hypergraph with $n$ vertices and minimum pair degree at least $(5/9+o(1))n^2/2$ contains a tight Hamiltonian cycle. This degree condition is asymptotically optimal.
We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of the original hypergraph. Accordingly we advocate their use in extremal hypergraph theory, and explain how they can lead to considerable simplifications in existing proofs in this field. We also use them for establishing the following two new results. Firstly, we prove a hypergraph extension of the Erdős-Gallai Theorem: for every $δ>0$ every sufficiently lar…
We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to Sudakov and Tomon, and our proof builds up on their work. A recent construction of B. Janzer implies that our bound is tight up to an $O((\log n)^4 \log \log n)$ factor.
We show that every 3-uniform hypergraph with $n$ vertices and minimum vertex degree at least $(5/9+o(1))\binom{n}2$ contains a tight Hamiltonian cycle. Known lower bound constructions show that this degree condition is asymptotically optimal.
A tight $k$-uniform $\ell$-cycle, denoted by $TC_\ell^k$, is a $k$-uniform hypergraph whose vertex set is $v_0, \cdots, v_{\ell-1}$, and the edges are all the $k$-tuples $\{v_i, v_{i+1}, \cdots, v_{i+k-1}\}$, with subscripts modulo $\ell$. Motivated by a classic result in graph theory that every $n$-vertex cycle-free graph has at most $n-1$ edges, Sós and, independently, Verstraëte asked whether for every integer $k$, a $k$-uniform $n$-vertex hypergraph without any tight $k$-uniform cycles has at most $\binom{n-1}{k-1}$ edges. In this paper, we answer this question in negative.
We prove that every 52-connected line graph of a rank 3 hypergraph is Hamiltonian. This is the first result of this type for hypergraphs of bounded rank other than ordinary graphs.
An $r$-uniform tight cycle of length $\ell>r$ is a hypergraph with vertices $v_1,\dots,v_\ell$ and edges $\{v_i,v_{i+1},\dots,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $\ell$. It was shown by Sudakov and Tomon that for each fixed $r\geq 3$, an $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle of any length has at most $n^{r-1+o(1)}$ hyperedges, but the best known construction (with the largest number of edges) only gives $Ω(n^{r-1})$ edges. In this note we prove that, for each fixed $r\geq 3$, there are $r$-uniform hypergraphs with $Ω(n^{r-1}\log n/\lo…
We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy considered it for tight Hamilton cycles for a fairly strong notion of uniformly dense hypergraphs. We focus on tight cycles and obtain optimal results for a weaker notion of uniformly dense hypergraphs. We show that if an $n$-vertex $3$-uniform hypergraph $H=(V,E)$ has the property that for any set of vertices $X$ and for any collection $P$ of pairs of vertices…
We give an algorithmic proof for the existence of tight Hamilton cycles in a random r-uniform hypergraph with edge probability p=n^{-1+eps} for every eps>0. This partly answers a question of Dudek and Frieze [Random Structures Algorithms], who used a second moment method to show that tight Hamilton cycles exist even for p=omega(n)/n (r>2) where omega(n) tends to infinity arbitrary slowly, and for p=(e+o(1))/n (r>3). The method we develop for proving our result applies to related problems as well.
We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) we have $|E_{i-1}\setminus E_i|=\ell$. We define a class of $(\e,p)$-regular hypergraphs, that includes random hypergraphs, for which we can prove the existence of a decomposition of almost all edges into type $\ell$ Hamilton cycles, where $\ell<k/2$.
Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-coloured complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for for all natural numbers $p$ and $r$, the vertices of every $r$-edge-coloured complete graph can be partitioned into a bounded number of $p$-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further …
We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Turán- and Dirac-type results. While the Turán-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields a sufficient condition relying solely on the minimum vertex degree.
An $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so that every edge $e$ contains a vertex $v$ that does not belong to any preceding edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai [Kalai], generalizing the Erdős-Sós Conjecture for trees, asserts that if $T$ is a tight $r$-tree with $t$ edges and $G$ is an $n$-vertex $r$-uniform hypergraph containing no copy of $T$ then $G$ has at most $\frac{t-1}{r}\binom{n}{r-1}$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree such that every edge of $T-T'$ has $r-1$ vertices in some edge of $T…
For integers $k \geq 3$ and $r\geq 2$, we show that for every $α> 0$, there exists $\varepsilon > 0$ such that the union of $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $αn$ and a binomial random $k$-uniform hypergraph $G^{(k)}(n,p)$ with $p\geq n^{-{\binom{k+r-2}{k-1}}^{-1}-\varepsilon}$ on the same vertex set contains the $r^{th}$ power of a tight Hamilton cycle with high probability. Moreover, a construction shows that one cannot take $\varepsilon > Cα$, where $C=C(k,r)$ is a constant. Thus the bound on $p$ is optimal up to the value of $\varepsilon$ and this answer…
The hypergraph regularity lemma -- the extension of Szemerédi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-Rödl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. In a recent paper we have shown that these bounds are unavoidable for $3$-uniform hypergraphs. In this paper we exten…
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop new techniques which enable us to prove that under certain natural pseudo-random conditions, almost all edges of H can be covered by edge-disjoint tight Hamilton cycles, for N divisible by 4. Consequently, we derive the corollary that random 3-uniform hypergraphs can be almost completely packed…
We show that, for a natural notion of quasirandomness in $k$-uniform hypergraphs, any quasirandom $k$-uniform hypergraph on $n$ vertices with constant edge density and minimum vertex degree $Ω(n^{k-1})$ contains a loose Hamilton cycle. We also give a construction to show that a $k$-uniform hypergraph satisfying these conditions need not contain a Hamilton $\ell$-cycle if $k-\ell$ divides $k$. The remaining values of $\ell$ form an interesting open question.
Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is best possible and the degree condition is asymptotically best possible. This result also has an interpretation as the \emph{deficiency problems}, recently introduced by Nenadov, Sudakov and Wagner: every such $H$ can be made Hamiltonian by adding at most two vertices and all triples intersecting them.
We consider a robust variant of Dirac-type problems in $k$-uniform hypergraphs. For instance, we prove that if $H$ is a $k$-uniform hypergraph with minimum codegree at least $(1/2 + γ)n$, $γ>0$, and $n$ is sufficiently large, then any edge coloring $ϕ$ satisfying appropriate local constraints yields a properly colored tight Hamilton cycle in $H$. Similar results for loose cycles are also shown.
We show that every $3$-uniform hypergraph $H=(V,E)$ with $|V(H)|=n$ and minimum pair degree at least $(4/5+o(1))n$ contains a squared Hamiltonian cycle. This may be regarded as a first step towards a hypergraph version of the Pósa-Seymour conjecture.
We prove a lower bound on the length of the longest $j$-tight cycle in a $k$-uniform binomial random hypergraph for any $2 \le j \le k-1$. We first prove the existence of a $j$-tight path of the required length. The standard "sprinkling" argument is not enough to show that this path can be closed to a $j$-tight cycle -- we therefore show that the path has many extensions, which is sufficient to allow the sprinkling to close the cycle.
We prove that for integers $2 \leq \ell < k$ and a small constant $c$, if a $k$-uniform hypergraph with linear minimum codegree is randomly `perturbed' by changing non-edges to edges independently at random with probability $p \geq O(n^{-(k-\ell)-c})$, then with high probability the resulting $k$-uniform hypergraph contains a Hamilton $\ell$-cycle. This complements a recent analogous result for Hamilton $1$-cycles due to Krivelevich, Kwan and Sudakov, and a comparable theorem in the graph case due to Bohman, Frieze and Martin.
We investigate minimum vertex degree conditions for $3$-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex. We prove that every $3$-uniform $n$-vertex ($n$ even) hypergraph $\mathcal{H}$ with minimum vertex degree $δ_1(\mathcal{H})\geq \left(\frac7{16}+o(1)\right)\binom{n}{2}$ contains a loose Hamilton cycle. This bound is asymptotically best possible.
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 $j<i$. Kalai conjectured in 1984 that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r}\binom{n}{r-1}$ edges contains every tight $r$-tree $T$ with $t$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree $T'$ of $T$ such that vertices in $V(T)\setminus V(T')$ are leaves in $T$. Kalai's Conjecture was proved in 1987 for tight $…
The hypergraph regularity lemma -- the extension of Szemerédi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-Rödl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $k \ge 2$, thus confirming a prediction of Tao. Pri…
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 prove that any k-uniform hypergraph on n vertices with minimum degree at least n/(2(k-1))+o(n) contains a loose Hamilton cycle. The proof strategy is similar to that used by Kühn and Osthus for the 3-uniform case. Though some additional difficulties arise in the k-uniform case, our argument here is considerably simplified by applying the recent hypergraph blow-up lemma of Keevash.
We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erdős and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.
In the random hypergraph $H_{n,p;k}$ each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle in which every pair of adjacent edges intersects in a single vertex. We prove that if $p n^{k-1}/\log n$ tends to infinity with $n$ then $$\lim_{\substack{n\to \infty 2(k-1) |n}}\Pr(H_{n,p;k}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$ This is asymptotically best possible.
In the random hypergraph H=H(n,p;3) each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges {x_i,y_i,x_{i+1}\} for i=1,2,...,n/2. We prove that there exists an absolute constant K>0 such that if p>K\log n/n^2 then lim_{n->oo 4 |n}}Pr(H(n,p;3) contains a loose Hamilton cycle)=1.
For $1\le d\le \ell< k$, we give a new lower bound for the minimum $d$-degree threshold that guarantees a Hamilton $\ell$-cycle in $k$-uniform hypergraphs. When $k\ge 4$ and $d< \ell=k-1$, this bound is larger than the conjectured minimum $d$-degree threshold for perfect matchings and thus disproves a well-known conjecture of Rödl and Ruciński. Our (simple) construction generalizes a construction of Katona and Kierstead and the space barrier for Hamilton cycles.
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 $k\geq n/2$.
We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so called {\em two-path} or {\em cherry}-quasirandom $3$-graphs. Our first result asserts that for any fixed real $α>0$, cherry-quasirandom $3$-graphs of sufficiently large order $n$ having minimum $2$-degree at least $α(n-2)$ have a tight Hamilton cycle. Our second result concerns the minimum $1$-degree sufficient for such $3$-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every $d,α>0$ satisfying $d + α>1$, any sufficiently large $n$-vertex suc…
We establish a precise characterisation of $4$-uniform hypergraphs with minimum codegree close to $n/2$ which contain a Hamilton $2$-cycle. As an immediate corollary we identify the exact Dirac threshold for Hamilton $2$-cycles in $4$-uniform hypergraphs. Moreover, by derandomising the proof of our characterisation we provide a polynomial-time algorithm which, given a $4$-uniform hypergraph $H$ with minimum codegree close to $n/2$, either finds a Hamilton $2$-cycle in $H$ or provides a certificate that no such cycle exists. This surprising result stands in contrast to the graph setting, in whi…
For $k\ge 2$ and $r\ge 1$ such that $k+r\ge 4$, we prove that, for any $α>0$, there exists $ε>0$ such that the union of an $n$-vertex $k$-graph with minimum codegree $\left(1-\binom{k+r-2}{k-1}^{-1}+α\right)n$ and a binomial random $k$-graph $\mathbb{G}^{(k)}(n,p)$ with $p\ge n^{-\binom{k+r-2}{k-1}^{-1}-ε}$ on the same vertex set contains the $r^{\text{th}}$ power of a tight Hamilton cycle with high probability. This result for $r=1$ was first proved by McDowell and Mycroft.
In an $r$-uniform hypergraph on $n$ vertices a tight Hamilton cycle consists of $n$ edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of $r$ vertices. We provide a first deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton cycles in random $r$-uniform hypergraphs with edge probability at least $C \log^3n/n$. Our result partially answers a question of Dudek and Frieze [Random Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton cycles exists already for $p=ω(1/n)$ for $r=3$ and $p=(e + o(…
We define and study a special type of hypergraph. A $σ$-hypergraph $H= H(n,r,q$ $\mid$ $σ$), where $σ$ is a partition of $r$, is an $r$-uniform hypergraph having $nq$ vertices partitioned into $ n$ classes of $q$ vertices each. If the classes are denoted by $V_1$, $V_2$,...,$V_n$, then a subset $K$ of $V(H)$ of size $r$ is an edge if the partition of $r$ formed by the non-zero cardinalities $ \mid$ $K$ $\cap$ $V_i \mid$, $ 1 \leq i \leq n$, is $σ$. The non-empty intersections $K$ $\cap$ $V_i$ are called the parts of $K$, and $s(σ)$ denotes the number of parts. We consider various types of cycl…
We show that every $k$-uniform hypergraph on $n$ vertices whose minimum $(k-2)$-degree is at least $(5/9+o(1))n^2/2$ contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and Sahueza-Matamala.
Given integers $k,j$ with $1\le j \le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the `Pathfinder' algorithm, a depth-first search algorithm which discovers $j$-tight paths in a $k$-uniform hypergraph. We prove that, in the supercritical case, w…
We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) we have $|E_{i-1}-E_i|=\ell$. We prove that for $\ell \le k\le 2\ell$, with high probability almost all edges of a random $k$-uniform hypergraph $H(n,p,k)$ with $p(n)\gg \log^2 n/n$ can be decomposed into edge disjoint type $\ell$ Hamilton cycles. We also provide sufficient condi…
The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer $N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$ Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to dete…
We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. We obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergraph with Omega(n^{k-epsilon}) edges, adding omega(n^{k epsilon/2}) random edges makes the hypergraph almost surely non-2-colorable. This is essentially tight, since there is a 2-colorable hypergraph with Omega(n^{k-ε}) edges which almost surely remains 2-colorable even after adding o(n^{k ε/ 2}) random edges.
For $1\le \ell<k/2$, we show that for sufficiently large $n$, every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\frac n{2 (k-\ell)} $ contains a Hamilton $\ell$-cycle. This codegree condition is best possible and improves on work of Hàn and Schacht who proved an asymptotic result.
We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$-uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal's theorem was taken by Pósa, who improved on Dirac's tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already …
For $s \ge 4$, the 3-uniform tight cycle $C^3_s$ has vertex set corresponding to $s$ distinct points on a circle and edge set given by the $s$ cyclic intervals of three consecutive points. For fixed $s \ge 4$ and $s \not\equiv 0$ (mod 3) we prove that there are positive constants $a$ and $b$ with $$2^{at}<r(C^3_s, K^3_t)<2^{bt^2\log t}.$$ The lower bound is obtained via a probabilistic construction. The upper bound for $s>5$ is proved by using supersaturation and the known upper bound for $r(K_4^{3}, K_t^3)$, while for $s=5$ it follows from a new upper bound for $r(K_5^{3-}, K_t^3)$ that we de…
We show that for sufficiently large $n$, every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $\binom{n-1}2 - \binom{\lfloor\frac34 n\rfloor}2 + c$, where $c=2$ if $n\in 4\mathbb{N}$ and $c=1$ if $n\in 2\mathbb{N}\setminus 4\mathbb{N}$, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Buß, Hàn and Schacht who proved the corresponding asymptotical result.
An {\em $\ell$-offset Hamilton cycle} $C$ in a $k$-uniform hypergraph $H$ on~$n$ vertices is a collection of edges of $H$ such that for some cyclic order of $[n]$ every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) satisfies $|E_{i-1}\cap E_i|=\ell$ and every pair of consecutive edges $E_{i},E_{i+1}$ in $C$ satisfies $|E_{i}\cap E_{i+1}|=k-\ell$. We show that in general $\sqrt{e^{k}\ell!(k-\ell)!/n^k}$ is the sharp threshold for the existence of the $\ell$-offset Hamilton cycle in the random $k$-uniform hypergraph $H_{n,p}^{(k)}$. We also examine this st…
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},…
