论文标题
简单同源物和痕量结合的超图
Simplicial homeomorphs and trace-bounded hypergraphs
论文作者
论文摘要
我们的第一个主要结果是在\ mathbb n $中的每个维度上，均匀限制了$ k $二维的简单综合体的拓扑数：对于\ mathbb n $中的每个$ k \，每个$λ_k\ ge k \ ge k^k^{ - 2k^2} $ for $ k $ k $ $ k $ compe} $ k $ - complex上的$ n \ ge n_0（\ nathcal {s}）$ vertices，至少$ n^{k+1 - λ_k} $ facets包含$ \ nathcal {s} $的同构副本。以前仅在一号和第二个方面都知道这一点，这都是由高度特定的参数：$λ_1$的存在是Mader的结果，是1967年以来的Mader的结果，Linial在2006年提出了$λ_2$的存在，最近由Keevash-Nong-Naryanan-Narayanan-Scott证明。 We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, \dots, V_r$ is said to be $d$-trace-bounded if for each $2 \le i \le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $ h $ on $ v_1 \ cup v_2 \ cup \ dots \ cup v_i $。我们的第二个主要结果是对Turán数量的量子量的以下估计值：对于所有$ r \ ge 2 $和$ d $和$ d \ in \ mathbb n $，$α_{r，d} \ ge（5rd）^{1-r} $在$ n \ ge n_0（h）上，至少$ n^{r -α_{r，d}} $ edges包含$ h $的副本。这加强了2009年的Conlon-Fox-Sudakov的结果，他表明，这种限制的限制为$ r $ r $ r $ r $ - graphs $ h $ $ h $，满足了更强有力的假设，即除了$ h $ h $，而不是在其轨迹中，除了$ h $ h $ h $ bundertite ofttex-degrees。
Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Turán numbers of $k$-dimensional simplicial complexes: for each $k \in \mathbb N$, there is a $λ_k \ge k^{-2k^2}$ such that for any $k$-complex $\mathcal{S}$, every $k$-complex on $n \ge n_0(\mathcal{S})$ vertices with at least $n^{k+1 - λ_k}$ facets contains a homeomorphic copy of $\mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $λ_1$ is a result of Mader from 1967, and the existence of $λ_2$ was suggested by Linial in 2006 and recently proved by Keevash-Long-Narayanan-Scott. We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, \dots, V_r$ is said to be $d$-trace-bounded if for each $2 \le i \le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 \cup V_2 \cup \dots \cup V_i$. Our second main result is the following estimate for the Turán numbers of degenerate trace-bounded hypergraphs: for all $r \ge 2$ and $d\in\mathbb N$, there is an $α_{r,d} \ge (5rd)^{1-r}$ such that for any $d$-trace-bounded $r$-partite $r$-graph $H$, every $r$-graph on $n \ge n_0(H)$ vertices with at least $n^{r - α_{r,d}}$ edges contains a copy of $H$. This strengthens a result of Conlon-Fox-Sudakov from 2009 who showed that such a bound holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).
