For fixed integers $r\ge 3,e\ge 3,v\ge r+1$, an $r$-uniform hypergraph is called $\mathscr{G}_r(v,e)$-free if the union of any $e$ distinct edges contains at least $v+1$ vertices. Brown, Erd\H{o}s and S\'{o}s showed that the maximum number of edges of such a hypergraph on $n$ vertices, denoted as $f_r(n,v,e)$, satisfies $$\Omega(n^{\frac{er-v}{e-1}})=f_r(n,v,e)=\mathcal{O}(n^{\lceil\frac{er-v}{e-1}\rceil}).$$ For $e-1\mid er-v$, the lower bound matches the upper bound up to a constant factor; whereas for $e-1\nmid er-v$, in general it is a notoriously hard problem to determine the correct exponent of $n$. Among other results, we improve the above lower bound by showing that $$f_r(n,v,e)=\Omega(n^{\frac{er-v}{e-1}}(\log n)^{\frac{1}{e-1}})$$ for any $r,e,v$ satisfying $\gcd(e-1,er-v)=1$. The hypergraph we constructed is in fact $\mathscr{G}_r(ir-\lceil\frac{(i-1)(er-v)}{e-1}\rceil,i)$-free for every $2\le i\le e$, and it has several interesting applications in Coding Theory. The proof of the new lower bound is based on a novel application of the lower bound on the hypergraph independence number due to Duke, Lefmann, and R{\"o}dl.

中文翻译：

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稀疏超图在编码理论中的应用

对于固定整数 $r\ge 3,e\ge 3,v\ge r+1$，$r$-uniform hypergraph 被称为 $\mathscr{G}_r(v,e)$-free 如果任何 $e$ 条不同的边至少包含 $v+1$ 个顶点。Brown、Erd\H{o}s 和 S\'{o}s 表明这种超图在 $n$ 个顶点上的最大边数，表示为 $f_r(n,v,e)$，满足 $$ \Omega(n^{\frac{er-v}{e-1}})=f_r(n,v,e)=\mathcal{O}(n^{\lceil\frac{er-v}{e -1}\rceil}).$$ 对于$e-1\mid er-v$，下限与上限匹配，直到一个常数因子；而对于 $e-1\nmid er-v$，通常来说，确定 $n$ 的正确指数是一个众所周知的难题。在其他结果中，我们通过证明 $$f_r(n,v,e)=\Omega(n^{\frac{er-v}{e-1}}(\log n)^{ \frac{1}{e-1}})$$ 对于满足 $\gcd(e-1,er-v)=1$ 的任何 $r,e,v$。我们构建的超图实际上是 $\mathscr{G}_r(ir-\lceil\frac{(i-1)(er-v)}{e-1}\rceil,i)$-free 每 $2\ le i\le e$，它在编码理论中有几个有趣的应用。新下界的证明基于基于杜克、莱夫曼和 R{\"o}dl 的超图独立数下界的新应用。