AI 中文总结
本文针对Turán $(r+1,r)$-系统的最小边数问题，通过概率构造改进了渐近界，还对构造去随机化并探讨了其在覆盖码中的应用。
AI 中文摘要
对于满足 $n\ge s>r$ 的正整数 $n$，令 $T(n,s,r)$ 表示 $n$ 个顶点的 $r$ 均匀超图中，使得每个 $s$ 顶点集至少包含一条边的最小边数。简单平均论证表明，比值 $T(n,s,r)/\binom nr$ 随 $n$ 非递减，记其 $n\to\infty$ 时的极限为 $t(s,r)$。当 $s=r+1$ 时该问题已有丰富研究历史，此前已知 $r\to\infty$ 时的渐近界为 $1\le r\cdot t(r+1,r)\le 4.91\dots$。本文提出一种简单概率构造，证明对所有 $r\ge1$，有 $(r+2)\cdot t(r+1,r)\le 4$；还对该构造进行了去随机化处理，并讨论了其在覆盖码中的应用。
英文摘要
For positive integers $n\ge s>r$, let $T(n,s,r)$ denote the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices such that every $s$-set of vertices contains at least one edge. A simple averaging argument shows that the ratio $T(n,s,r)/\binom nr$ is non-decreasing in $n$ and we denote its limit as $n\to\infty$ by $t(s,r)$. The case $s=r+1$ has a rich history, with the previously best known asymptotic bounds for $r\to\infty$ being $1\le r\cdot t(r+1,r)\le 4.91...$ . In this paper, we present a simple probabilistic construction which shows that $(r+2)\cdot t(r+1,r)\le 4$ for every $r\ge1$. We also derandomise it and discuss applications to covering codes.
