How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given r-edge-coloured graph G? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph H if it is known that any collection of a few edges of H has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Cirão, Letzter and Sahasrabudhe.

一个给定r边色图G的所有顶点需要覆盖多少个单色路径，循环或一般树？这些问题是在1960年代引入的，在过去的50年中，许多研究人员对其进行了深入的研究。在本文中，我们在超图中建立了该问题与以下自然Helly型问题之间的联系。粗略地说，这个问题要求覆盖超图H的所有边缘所需的最大顶点数，如果已知H的一些边缘的任何集合有一个小的封面。我们获得了关于超图问题的非常准确的界线，并使用它们来给出一些意外的答案，这些问题涉及由Bal和DeBiasio，Kohayakawa，Mota和Schacht，Lang和Lo以及Cirão，Litzter和Sahasrabudhe提出和研究的单色树覆盖图。