Abstract: A conflict-free colourings of t-subsets in hypergraphs is a colouring where one assigns colours to all subsets of vertices of cardinality t such that in any hyperedge of cardinality at least t there is a uniquely coloured t-subset. In the talk, I will present the following results.
(i) For any fixed t, the t-subsets in any set P of n points in the plane can be coloured with O(t2 log2 n) colours so that any axis-parallel rectangle that contains at least t points of P also contains a uniquely coloured t-subset.
(ii) For a wide class of `well behaved' geometrically defined hypergraphs, there is a nearly tight upper bound on their t-subset conflict-free chromatic number.
This is a joint work with Bruno Jartoux, Chaya Keller and Shakhar Smorodinsky.