AI 中文总结
本文提出一个将禁用迹的固定一致性界转化为对一致性多项式依赖界的一般原理，并给出迹-$C_4$-自由超图的上下界，上界为$O(r^{3/2+\varepsilon}m^{3/2})$，下界为$\Omega(r^{1/2}m^{3/2})$。
AI 中文摘要
我们给出一个一般原理，将禁用迹的固定一致性界转化为对一致性具有多项式依赖的界。更精确地说，设$H$是$h\ge1$个顶点上的固定集族，并假设对于某个$\alpha\ge0$，对每个固定整数$j\ge1$都有$\operatorname{ex}_j(m,\operatorname{Tr}(H))=O_{H,j}(m^\alpha)$。那么，对每个$\varepsilon>0$，存在常数$C_{H,\alpha,\varepsilon}$使得对所有$m\ge r\ge2$有\\[ \operatorname{ex}_r(m,\operatorname{Tr}(H)) \le C_{H,\alpha,\varepsilon} r^{h-1-\alpha+\varepsilon}m^\alpha。\\]特别地，对于迹-$C_4$-自由超图，对每个$\varepsilon>0$存在常数$C_\varepsilon$使得对所有$m\ge r\ge2$有\\[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \le C_\varepsilon r^{3/2+\varepsilon}m^{3/2}。\\]我们还构造了迹-$C_4$-自由的$r$-图，证明了对绝对常数$c>0$，对每个固定$r\ge3$和所有足够大的$m$（依赖于$r$）有\\[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \ge c r^{1/2}m^{3/2}。\\]
英文摘要
We give a general principle that converts fixed-uniformity bounds for forbidden traces into bounds with polynomial dependence on the uniformity. More precisely, let $H$ be a fixed set system on $h\ge1$ vertices, and suppose that, for some $α\ge0$, $\operatorname{ex}_j(m,\operatorname{Tr}(H))=O_{H,j}(m^α)$ for every fixed integer $j\ge1$. Then, for every $\varepsilon>0$, there is a constant $C_{H,α,\varepsilon}$ such that \[ \operatorname{ex}_r(m,\operatorname{Tr}(H)) \le C_{H,α,\varepsilon} r^{h-1-α+\varepsilon}m^α\] for all $m\ge r\ge2$. In particular, for trace-$C_4$-free hypergraphs and every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that \[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \le C_\varepsilon r^{3/2+\varepsilon}m^{3/2} \] for all $m\ge r\ge2$. We also construct trace-$C_4$-free $r$-graphs showing that \[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \ge c r^{1/2}m^{3/2} \] for an absolute constant $c>0$, for every fixed $r\ge3$ and all sufficiently large $m$ (depending on $r$).
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