A further generalization of the colourful Carath\'eodory theorem

Given $d+1$ sets, or colours, $S_1,S_2,\mathrm{.}\mathrm{.}\mathrm{.},S_{d+1}$ of points in ${\mathbb{R}}^d$, a {\em colourful} set is a set $S\subseteq {\bigcup }_i{S}_i$ such that $\mathrm{\mid }S\cap S_i\mathrm{\mid }\le 1$ for $i=1,\mathrm{.}\mathrm{.}\mathrm{.},d+1$. The convex hull of a colourful set $S$ is called a {\em colourful simplex}. B'ar'any’s colourful Carath'eodory theorem asserts that if the origin 0 is contained in the convex hull of $S_i$ for $i=1,\mathrm{.}\mathrm{.}\mathrm{.},d+1$, then there exists a colourful simplex containing 0. The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of $S_i\cup S_j$ for $1\leq i< j \leq d+1$ by Arocha et al. and by Holmsen et al. We further generalize the sufficient condition and obtain new colourful Carath'eodory theorems. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternative, and more general, proof using graphs is given. In addition, we observe that any condition implying the existence of a colourful simplex containing 0 actually implies the existence of ${\min }_i\mathrm{\mid }{S}_i\mathrm{\mid }$ such simplices.