We investigate here the behaviour of a large typical meandric system, proving a central limit theorem for the number of components of a given shape. Our main tool is a theorem of Gao and Wormald that allows us to deduce a central limit theorem from the asymptotics of large moments of our quantities of interest.
We show that for every 
$eta gt 0$ every sufficiently large 
$n$-vertex oriented graph 
$D$ of minimum semidegree exceeding 
$(1+eta )frac k2$ contains every balanced antidirected tree with 
$k$ edges and bounded maximum degree, if 
$kge eta n$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs.
Further, we show that in the same setting, 
$D$ contains every 
$k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length 
For a graph 
$H$ and a hypercube 
$Q_n$, 
$textrm{ex}(Q_n, H)$ is the largest number of edges in an 
$H$-free subgraph of 
$Q_n$. If 
$lim _{n rightarrow infty } textrm{ex}(Q_n, H)/|E(Q_n)| gt 0$, 
$H$ is said to have a positive Turán density in a hypercube or simply a positive Turán density; otherwise, it has zero Turán density. Determining 
$textrm{ex}(Q_n, H)$ and even identifying whether 
$H$ has a positive or zero Turán density remains a widely open question for general
Tao and Vu showed that every centrally symmetric convex progression 
$Csubset mathbb{Z}^d$ is contained in a generalized arithmetic progression of size 
$d^{O(d^2)} # C$. Berg and Henk improved the size bound to 
$d^{O(dlog d)} # C$. We obtain the bound 
$d^{O(d)} # C$, which is sharp up to the implied constant and is of the same form as the bound in the continuous setting given by John’s theorem.
A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary 
$r$-colouring of the complete 
$k$-uniform hypergraph 
$K_n^k$ when 
$kgeq 2$ and 
$kin {r-1,r}$. We prove a result which says that if one replaces 
$K_n^k$ in Gyárfás’ theorem by any ‘expansive’ 
$k$-uniform hypergraph on 
$n$ vertices (that is, a 
$k$-uniform hypergraph 
Gyárfás [12] 的一个结果精确地确定了当 $kgeq 2$ 和 $kin{r-1,r}$ 时，完整 $k$ Uniform 超图 $K_n^k$ 的任意 $r$ 着色中最大单色分量的大小。我们证明了这样一个结果：如果把 Gyárfás 定理中的 $K_n^k$ 替换为任何在 $n$ 顶点上的 "扩张性"$k$-匀速超图（即在 $n$ 顶点上的 $k$ 匀速超图 $G$，其中 $e(V_1, ldots、V_k)gt 0$ 对于所有不相邻的集合 $V_1, ldots, V_ksubseteq V(G)$，对于 [k]$ 中的所有 $i 都是 $|V_i|gtalpha$），那么我们会得到一个大小基本相同的最大单色分量（在取决于 $r$ 和 $alpha$ 的小误差范围内）。Gyárfás 的结果等同于一个对偶问题，即确定一个具有 $n$ 边的任意 $r$ 部分 $r$-Uniform 超图 $H$ 的最小最大度，其中每组 $k$ 边都有一个共同交集。用这种语言表达，我们的结果就是说，如果把每组 $k$ 边有共同交集的条件换成这样的条件：对于每组 $k$ 不相交的集合 $E_1, ldots, E_ksubseteq E(H)$ 带有 $|E_i|gt alpha$、存在 $(e_1, ldots, e_k)in E_1times cdots times E_k$，使得 $e_1cap cdots cap e_kneq emptyset$，那么 $H$ 的最小可能最大度基本上是相同的（在取决于 $r$ 和 $alpha$ 的小误差范围内）。我们将在这种对偶设置中证明我们的结果。
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