Let P be a set of 2n points in convex position, such that n points are colored red and n points are colored blue. A non-crossing alternating path on P of length l is a sequence p_1, ..., p_l of l points from P so that (i) all points are pairwise distinct; (ii) any two consecutive points p_i, p_{i+1} have different colors; and (iii) any two segments p_i p_{i+1} and p_j p_{j+1} have disjoint relative interiors,
for i /= j.

We show that there is an absolute constant eps > 0, independent of n and of the coloring, such that P always admits a non-crossing alternating path of length at least (1 + eps)n. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least (1 + eps)n points of P. This is a properly colored matching whose segments are pairwise disjoint and intersected by a common line. For both versions, this is the first improvement of the easily obtained lower bound of n by an additive term linear in n. The best known published upper bounds are asymptotically of order 4n/3 + o(n).

Based on joint work with Pavel Valtr.