随机独立集在无三角形和线性Berge-$C_4$-自由超图中的应用
Random independent sets in triangle-free and linear Berge-$C_4$-free hypergraphs
AI总结：
研究无三角形和线性Berge-$C_4$-自由超图中随机独立集的存在性，提出基于有序随机选取过程的方法，给出顶点包含概率下界及分数色数上界。
AI中文摘要：
我们研究了具有局部约束的一致超图独立集上的概率分布。对于$(r+1)$-一致超图上的正顶点权重函数$w$，定义顶点$v$的加权度为\\[ d_w(v):= \sum_{e\ni v} \left( \prod_{u\in e\setminus\{v\}} \dfrac{w(u)}{w(v)} \right)^{1/r}. \\] 对于每个固定的$r\geq1$，我们的第一个结果给出，在每个无三角形（不要求线性）的$(r+1)$-一致超图中，存在一个随机独立集$I$满足\\[ \mathbb P(v\in I)\geq \left( \frac{r}{r+1}(r(r+1))^{-1/r}+o_r(1) \right) \left(\frac{\log d_w(v)}{d_w(v)}\right)^{1/r}, \qquad d_w(v)\to+\infty. \\] 同样的结果也蕴含每个$d$-退化的无三角形$(r+1)$-一致超图满足\\[ \chi_f(G)\leq \left( \dfrac{r+1}{r}(r(r+1))^{1/r}+o_r(1) \right) \left(\dfrac{d}{\log d}\right)^{1/r}, \qquad d\to+\infty. \\] 我们的第二个结果涉及允许Berge三角形的线性Berge-$C_4$-自由超图。在此设置下，对于每个足够大的阈值$D$，存在一个随机独立集$I$，使得对所有满足$d_w(v)\geq D$的顶点一致地有\\[ \mathbb P(v\in I)\geq (1-o_r(1)) \left(\frac{\log d_w(v)}{r\\,d_w(v)}\right)^{1/r}, \qquad D\to+\infty. \\] 这也得到\\[ \chi_f(G)\leq (1+o_r(1)) \left(\frac{rd}{\log d}\right)^{1/r} \\] 对于$d$-退化的线性Berge-$C_4$-自由$(r+1)$-一致超图。两个结果都基于源自Martinsson和Steiner的有序随机选取过程。通过选择不同的选择函数和迭代，我们证明该随机过程的输出给出了所需的随机独立集。
英文摘要：
We study probability distributions on independent sets of uniform hypergraphs with local constraints. For a positive vertex-weight function $w$ on an $(r+1)$-uniform hypergraph, define the weighted degree of vertex $v$ by \[ d_w(v):= \sum_{e\ni v} \left( \prod_{u\in e\setminus\{v\}}\dfrac{w(u)}{w(v)} \right)^{1/r}. \] For each fixed $r\geq1$, our first result gives, in every triangle-free $(r+1)$-uniform hypergraph (without a linearity assumption), a random independent set $I$ satisfying \[ \mathbb P(v\in I)\geq \left( \frac{r}{r+1}(r(r+1))^{-1/r}+o_r(1) \right) \left(\frac{\log d_w(v)}{d_w(v)}\right)^{1/r}, \qquad d_w(v)\to+\infty. \] The same result also implies that every $d$-degenerate triangle-free $(r+1)$-uniform hypergraph satisfies \[ χ_f(G)\leq \left( \dfrac{r+1}{r}(r(r+1))^{1/r}+o_r(1) \right) \left(\dfrac{d}{\log d}\right)^{1/r}, \qquad d\to+\infty. \] Our second result concerns linear Berge-$C_4$-free hypergraphs which allow Berge triangles. In this setting, for every sufficiently large threshold $D$, there exists a random independent set $I$ such that, uniformly over all vertices with $d_w(v)\geq D$, \[ \mathbb P(v\in I)\geq (1-o_r(1)) \left(\frac{\log d_w(v)}{r\,d_w(v)}\right)^{1/r}, \qquad D\to+\infty. \] It also yields \[ χ_f(G)\leq (1+o_r(1)) \left(\frac{rd}{\log d}\right)^{1/r} \] for $d$-degenerate linear Berge-$C_4$-free $(r+1)$-uniform hypergraphs. Both results are based on the ordered random pick process originated from Martinsson and Steiner. By choosing different selection functions and iteration, we prove the output of the random process gives the required random independent set.
