论文标题
确切的线性帆的线性数量
The exact linear Turán number of the Sail
论文作者
论文摘要
如果其两个边缘在一个顶点中相交，则超图是线性的。帆（或$ 3 $ -FAN）$ f^3 $是$ 3 $均匀的线性线性超图，由$ 3 $ edges $ f_1，f_1，f_2，f_3 $ pairywise在同一顶点$ v $中相交，额外的edge $ g $与$ v $不同的顶点不同的$ f_i $。线性turán数字$ ex_ {lin}（n，f^3）$是$ 3 $ - 均匀的线性超图中的最大边数，$ n $ vertices上不包含$ f^3 $的副本。 füredi和gyárfás证明，如果$ n = 3k $，则$ ex_ {lin}（n，f^3）= k^2 $，在这种情况下，唯一的极端超图是横向设计。他们还表明，如果$ n = 3k+2 $，则$ ex_ {lin}（n，f^3）= k^2+k $，而唯一的极端超图是截断的设计（从$ 3K+3 $的横向设计中获得了$ 3K+3 $ VERTICES的横向设计，并通过$ 3 $组以及所有超级词以及所有超级效果以及三个超过三个超过三个超过三个hyperepers and there hypergraphs and ytry ands hypere）。但是，当$ n = 3k+1 $ $打开时的情况。 在本文中，我们通过证明$ ex_ {lin}（n，f^3）= k^2+1 $如果$ n = 3k+1 $，回答了füredi和gyárfás的问题，来解决剩余的情况。我们还表征了所有极端超图。这种情况的困难是由于这些极端例子是不合时宜的。特别是，它们不是像其他情况一样来自横向设计。
A hypergraph is linear if any two of its edges intersect in at most one vertex. The Sail (or $3$-fan) $F^3$ is the $3$-uniform linear hypergraph consisting of $3$ edges $f_1, f_2, f_3$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. The linear Turán number $ex_{lin}(n, F^3)$ is the maximum number of edges in a $3$-uniform linear hypergraph on $n$ vertices that does not contain a copy of $F^3$. Füredi and Gyárfás proved that if $n = 3k$, then $ex_{lin}(n, F^3) = k^2$ and the only extremal hypergraphs in this case are transversal designs. They also showed that if $n = 3k+2$, then $ex_{lin}(n, F^3) = k^2+k$, and the only extremal hypergraphs are truncated designs (which are obtained from a transversal design on $3k+3$ vertices with $3$ groups by removing one vertex and all the hyperedges containing it) along with three other small hypergraphs. However, the case when $n =3k+1$ was left open. In this paper, we solve this remaining case by proving that $ex_{lin}(n, F^3) = k^2+1$ if $n = 3k+1$, answering a question of Füredi and Gyárfás. We also characterize all the extremal hypergraphs. The difficulty of this case is due to the fact that these extremal examples are rather non-standard. In particular, they are not derived from transversal designs like in the other cases.
