An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belongs to at most one edge. A linear $r$-graph is complete if every pair of vertices is in exactly one edge. The famous Brown–Erdős–Sós conjecture states that for every fixed $k$ and $r$, every linear $r$-graph with $\mathrm{\Omega }(n^2)$ edges contains $k$ edges spanned by at most $(r-2)k+3$ vertices. As an intermediate step towards this conjecture, Conlon and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely, that for every fixed $k$, $r$ and $c$, in every $c$-colouring of a complete linear $r$-graph, one can find $k$ monochromatic edges spanned by at most $(r-2)k+3$ vertices. We prove that this Ramsey version of the conjecture holds under the additional assumption that $r⩾r_0(c)$, and we show that for $c=2$ it holds for all $r⩾4$.

中文翻译：

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Brown-Erdős-Sós 猜想的拉姆齐变体

一个 $r$-均匀超图（$r$-graph 简称）如果每对顶点最多属于一个边，则称为线性。一个线性$r$如果每对顶点恰好在一条边上，则 -graph 是完整的。著名的 Brown-Erdős-Sós 猜想指出，对于每个固定的$克$ 和 $r$, 每个线性 $r$-图形与 $\mathrm{\Omega }(n^2)$ 边包含 $克$ 最多跨越的边 $(r-2)克+3$顶点。作为实现这一猜想的中间步骤，Conlon 和 Nenadov 最近建议证明其自然的 Ramsey 松弛。也就是说，对于每一个固定$克$, $r$ 和 $C$， 在每一个 $C$- 完整线性的着色 $r$-graph，可以找到 $克$ 最多跨越的单色边缘 $(r-2)克+3$顶点。我们证明该猜想的拉姆齐版本在附加假设下成立$r⩾r_0(C)$，我们证明对于 $C=2$ 它适用于所有人 $r⩾4$.