SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1493-1504, January 2020.
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, Erdös, and Sós showed that the maximum number of edges of such a hypergraph on $n$ vertices, denoted as $f_r(n,v,e)$, satisfies $\Omega(n^{\frac{er-v}{e-1}})=f_r(n,v,e)=O(n^{\lceil\frac{er-v}{e-1}\rceil})$. For sufficiently large $n$ and $e-1\mid er-v$, the lower bound matches the upper bound up to a constant factor, which depends only on $r,v,e$; 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 Rödl.

中文翻译：

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稀疏超图及其在编码理论中的应用

SIAM离散数学杂志，第34卷，第3期，第1493-1504页，2020年1月。
对于固定整数$ r \ ge 3，e \ ge 3，v \ ge r + 1 $，一个统一的$ r $超图称为$ \ mathscr {G} _r（v，e）$-，如果$ e $个不同的边至少包含$ v + 1 $个顶点。Brown，Erdös和Sós表明，这样的超图在$ 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）= O（n ^ {\ lceil \ frac {er-v} {e-1} \ rceil}）$。对于足够大的$ n $和$ e-1 \ mid er-v $，下限与上限匹配一个恒定因子，该常数仅取决于$ r，v，e $；而对于$ e-1 \ nmid er-v $，通常很难确定$ n $的指数。在其他结果中，我们通过显示$ f_r（n，v，e）= \ Omega（n ^ {\ frac {er-v} {e-1}}（\ log n）^ {\ frac {1} {e-1}}）$满足$ \ gcd（e-1，er-v）= 1 $的任何$ r，e，v $。实际上，我们构造的超图是$ \ mathscr {G} _r（ir- \ lceil \ frac {（i-1）（er-v）} {e-1} \ rceil，i），每$ 2免费文件，它在编码理论中有许多有趣的应用。新下界的证明是基于杜克，莱夫曼和罗德尔所带来的超图独立数下界的新颖应用。