论文标题
新的Turán指数，用于两个极端超毛皮问题
New Turán exponents for two extremal hypergraph problems
论文作者
论文摘要
如果任何$ t+2 $不同的边缘，$ r $ - 均匀的超图称为$ t $ cancellative $ a_1，\ ldots，a_t，b，c $，它认为$（\ cup_ {\ cup_ {i = 1}^t a_i）\ cup b \ neq b \ neq（\ cup_ \ cup_ {\ cup_ {i = 1}^t a_i）如果对于任何两个不同的子集$ \ MATHCAL {a}，\ MATHCAL {B} $，则称为$ t $ - union-fim-fim-form-union-fim-union，每个都包含$ \ cup_ {a \ in \ mathcal {a} a} a \ neq \ neq \ cup_ b \ b \ b \ b $ b \ b $ b \ b \ b \ b \ b \ b \ b \ b $令$ c_t（n，r）$（分别$ u_t（n，r）$）表示$ t $ cancellative的最大边数（resp。$ t $ tumion-free）$ r $ r $ -r $均匀的超透明图。除其他结果外，我们表明，对于固定$ r \ ge 3，t \ ge 3 $和$ n \ rightarrow \ infty $ $$ω（n^{\ lfloor \ frac {2r} {t+2} \ rfloor+\ frac {2r \ pmod {t+2}} {t+1}} {t+1}}}}} = c_t（n^，r） t/2 \ rfloor+1} \ rceil}）\ text {and}ω（n^{\ frac {\ frac {r} {t-1}}）= u_t（n，r）= o（n^{n^{\ lceil \ lceil \ frac \ frac {r}边界。特别是，当$ 2 \ mid t \ text {and}（t/2+1）\ mid r $和$ u_t（n，r）$时，当$ 2 \ text {and} $ 2 \ text {and}时，我们确定$ c_t（n，r）$的turán指数。 证明这两个下限的主要工具是这些问题与稀疏超图之间的新联系。
An $r$-uniform hypergraph is called $t$-cancellative if for any $t+2$ distinct edges $A_1,\ldots,A_t,B,C$, it holds that $(\cup_{i=1}^t A_i)\cup B\neq (\cup_{i=1}^t A_i)\cup C$. It is called $t$-union-free if for any two distinct subsets $\mathcal{A},\mathcal{B}$, each consisting of at most $t$ edges, it holds that $\cup_{A\in\mathcal{A}} A\neq \cup_{B\in\mathcal{B}} B$. Let $C_t(n,r)$ (resp. $U_t(n,r)$) denote the maximum number of edges of a $t$-cancellative (resp. $t$-union-free) $r$-uniform hypergraph on $n$ vertices. Among other results, we show that for fixed $r\ge 3,t\ge 3$ and $n\rightarrow\infty$ $$Ω(n^{\lfloor\frac{2r}{t+2}\rfloor+\frac{2r\pmod{t+2}}{t+1}})=C_t(n,r)=O(n^{\lceil\frac{r}{\lfloor t/2\rfloor+1}\rceil})\text{ and } Ω(n^{\frac{r}{t-1}})=U_t(n,r)=O(n^{\lceil\frac{r}{t-1}\rceil}),$$ thereby significantly narrowing the gap between the previously known lower and upper bounds. In particular, we determine the Turán exponent of $C_t(n,r)$ when $2\mid t \text{ and } (t/2+1)\mid r$, and of $U_t(n,r)$ when $(t-1)\mid r$. The main tool used in proving the two lower bounds is a novel connection between these problems and sparse hypergraphs.
