For a positive integer k , we say that a graph is k -existentially complete if for every 0 ⩽ a ⩽ k , and every tuple of distinct vertices x 1 , …, x a , y 1 , …, y k−a , there exists a vertex z that is joined to all of the vertices x 1 , …, x a and to none of the vertices y 1 , …, y k−a . While it is easy to show that the binomial random graph G n ,1/2 satisfies this property (with high probability) for k = (1 − o (1)) log 2 n , little is known about the “triangle-free” version of this problem: does there exist a finite triangle-free graph G with a similar “extension property”? This question was first raised by Cherlin in 1993 and remains open even in the case k = 4. We show that there are no k -existentially complete triangle-free graphs on n vertices with $$k\, > \,{{8\log \,n} \over {\log \,\log n}}$$ k > 8 log n log log n , for n sufficiently large.

中文翻译：

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关于存在完全无三角形图

对于正整数 k ，如果对于每个 0 ⩽ a ⩽ k 和每个不同顶点的元组 x 1 , …, xa , y 1 , …, yk−a ，存在一个顶点 z 与所有顶点 x 1 , ..., xa 相连，但不与任何顶点 y 1 , ..., yk−a 相连。虽然很容易证明二项式随机图 G n ,1/2 满足 k = (1 − o (1)) log 2 n 的这个性质（概率很高），但对“无三角形”知之甚少此问题的版本：是否存在具有类似“扩展属性”的有限三角形无图 G？这个问题由 Cherlin 于 1993 年首次提出，即使在 k = 4 的情况下仍然存在。我们证明了在 n 个顶点上不存在 k 存在完全无三角形的图，其中 $$k\, > \,{{8\ log \,n} \over {\log \,\log n}}$$ k > 8 log n log log n ,