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The complexity of two related geometrical (indeed, combinatorial) problems isconsidered, measured by the number of queries needed to determine the solution.It is proved that one cannot check in a linear in N number of queries whether Nintervals cover a whole interval, or whether for N pairs of intervals on twolines there is a pair of points intersecting each of these pairs of intervals("piercing all pairs of intervals"). The proofs are related to examples whichshow that there is no "Helly property" here - the whole set of N may cover thewhole interval (resp. may have no pair of points piercing all pairs ofintervals) while any proper subset does not. Also, for the piercing problem weoutline an algorithm, taking O(N log N) steps, to check whether there is a pairof points piercing all pairs of intervals and if there is, to find it.

No O(N) queries for checking if N intervals cover everything or for piercing N pairs of intervals. An O(N log N)-steps algorithm for piercing