AI 中文总结
该文研究k-图中Helly型族与无三角形族的稳定性，在更宽松的n>2k条件下证明了非星型Helly族的大小上界，并推导了无三角形族对应上界的成立与不成立的参数范围。
AI 中文摘要
我们研究k-图$\u200b\f{F}\u200b\binom{[n]}{k}$，其中$k\bgeq 3$。若一个k-图的任意两条边都有非空交集，则称其为相交的；若所有边都共享一个公共顶点，则称其为星型的。若k-图$\f{F}$的所有相交子族都是星型的，则称其为Helly的；若仅要求由三条边构成的子族满足该性质，则称其为无三角形的。众所周知，当$n\bgeq 3k/2$时，完全星是唯一最大的无三角形族，因此也是最大的Helly族。1984年Tuza证明了非星型Helly族的最优上界$|\f{F}|\bleq \binom{n-k-1}{k-1}+\binom{n-2}{k-2}+1$，但仅对某个未明确给出的$n>n_0(k)$成立。\n本文的研究目标有两个。首先，我们在$n>2k$的条件下建立了相同的上界。其次，我们证明当$n>12k^2$时，该上界对无三角形族同样成立；同时我们也证明，当$2k<n\bleq 3k-4$时该结论不成立。
英文摘要
We consider $k$-graphs, $\mathcal{F}\subset \binom{[n]}{k}$, $k\geq 3$. A $k$-graph is called intersecting if any two of its edges have non-empty intersection. It is called a star if all its edges share a common vertex. The $k$-graph $\mathcal{F}$ is called Helly if all its intersecting subfamilies are stars. If the same is required only for subfamilies consisting of three edges, it is called triangle-free. It is well known that for $n\geq 3k/2$, the full star is the unique largest triangle-free family whence the largest Helly family as well. In 1984 Tuza proved the best possible bound $|\mathcal{F}|\leq \binom{n-k-1}{k-1}+\binom{n-2}{k-2}+1$ for Helly families that are not stars, albeit only for some unspecified $n>n_0(k)$. The aim of this paper is twofold. First we establish the same bound for $n>2k$. Second we show that for $n>12k^2$ the same upper bound holds for triangle-free families. It is shown as well that it is not true for $2k<n\leq 3k-4$.
