AI 中文总结
研究3 - 稳定克内泽尔图的色数，证明了梅尼耶猜想在\(s = 3\)且\(n\)足够大或\(k = s = 3\)时成立，通过证明\(s\) - 稳定集的希尔顿 - 米尔纳定理版本及采用拓扑方法实现。
AI 中文摘要
对于整数\(s \geq 2\)，若子集\(S \subseteq [n]\)满足对任意\(i,j \in S\)且\(i < j\)，有\(\min \{j - i, n + i - j\}\geq s\)，则称\(S\)为\(s\) - 稳定的。用\(\binom{[n]}{k}_{s\text{-stable}}\)表示\([n]\)中所有大小为\(k\)的\(s\) - 稳定子集的集合。1978年施里弗证明，当\(n\geq 2k\)时，克内泽尔图\(\mathrm{KG}\big( \binom{[n]}{k}_{2\text{-stable}}\big)\)的色数为\(n - 2k +2\)。2项式猜想。2011年梅尼耶猜想对于所有\(n\geq sk\)，\(\chi\left( \mathrm{KG}\big( \binom{[n]}{k}_{s\text{-stable}} \big) \right)= n - sk +s\)。此前该猜想在\(s\)为偶数、\(s \geq 4\)且\(n\)足够大以及\(k = 2\)的情况下已被证明。我们证明了在\(s = 3\)且\(n\)足够大或\(k = s = 3\)的情况下该猜想成立。为此，我们证明了\(s\) - 稳定集的希尔顿 - 米尔纳定理的版本。我们还提出了一种针对梅尼耶猜想的拓扑方法。
英文摘要
For an integer $s \ge 2$, a subset $S \subseteq [n]$ is {\em $s$-stable} if $\min \{j - i, n + i - j\}\ge s$ for every $i,j \in S$ with $i<j$. Denote the set of all $s$-stable subsets of size $k$ of $[n]$ by $\binom{[n]}{k}_{s\text{-stable}}$. Schrijver proved in 1978 that whenever $n\ge 2k$, the chromatic number of the Kneser graph $\mathrm{KG}\big( \binom{[n]}{k}_{2\text{-stable}}\big)$ is $n - 2k +2$. Generalizing this result, Meunier conjectured in 2011 that $χ\left( \mathrm{KG}\big( \binom{[n]}{k}_{s\text{-stable}} \big) \right)= n - sk +s$ for all $n\ge sk$. This conjecture was previously proven for all even $s$, for $s \ge 4$ and large enough $n$, and for $k=2$. We prove the conjecture in the cases $s=3$ and $n$ large enough, or $k=s=3$. To this end, we prove versions of the Hilton-Milner theorem for $s$-stable sets. We also present a topological approach towards Meunier's conjecture.
