A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$ , then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour.Our result is tight in the sense that no longer cycles (of length $>2t+r$ ) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$ -vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$ , in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$ .

中文翻译：

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具有大最小度的 2 边彩色图中的单色路径和循环

图表G 箭头图表H如果在每个 2 边缘着色G存在一个单色副本H. Schhelp 的想法是，如果完整的图$K_n$箭头小图H，然后每个“密集”子图$K_n$还有箭头H，他概述了这个方向的一些问题。我们的主要成果就是本着这种精神。我们证明对于每一个足够大的n， 如果$n = 3t+r$在哪里$r \in \{0,1,2\}$和G是一个n-顶点图$\delta(G) \ge (3n-1)/4$，然后对于每 2 边着色G, 要么有每个长度的循环$\{3, 4, 5, \dots, 2t+r\}$相同的颜色，或者有每个偶数长度的循环$\{4, 6, 8, \dots, 2t+2\}$相同颜色的。我们的结果是紧密的，因为不再循环（长度$>2t+r$) 可以保证，不能降低最小度条件。这也暗示了 Schhelp 的猜想，对于每一个足够大的n， 每一个$(3t-1)$-顶点图G最小度数大于$3|V(G)|/4$箭头路径$P_{2n}$有 2n顶点。此外，它意味着足够大nBenevides、Łuczak、Scott、Skokan 和 White 的猜想$n=3t+r$在哪里$r \in \{0,1,2\}$和每一个n-顶点图 G 与$\delta(G) \ge 3n/4$，在每个 2 边着色中G至少存在一个长度的单色循环$2t+r$.