Given r families of subsets of a fixed n-set, we say that they are r-cross t-intersecting if for every choice of representatives, exactly one from each family, the common intersection of these representatives is of size at least t. We obtain a generalisation of a result by Hilton and Milner on cross intersecting families. In particular, we determine the maximum possible sum of the sizes of non-empty r-cross t-intersecting families in the case when all families are k-uniform and in the case when they are arbitrary subfamilies of the power set. Only some special cases of these results had been proved before. The method we use also yields more general results concerning measures of families instead of their sizes.

We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-colored random graph. More precisely, we show that for $p\gg n^{-1/6}{(\ln n)}^{1/6}$, in any $3$-edge coloring of the random graph $G(n,p)$ we can find three monochromatic trees such that their union covers all vertices. This improves, for three colors, a result of Bucić, Korándi, and Sudakov.