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We have a circle and two families of $n$ red arcs and $n$ blue arcs, positioned on the circle so that every two arcs of different colors intersect. Can one show that there is a point in the perimeter which is part of at least $n$ arcs?

(The statement sounds very simple. It makes me think the answer should be very simple too, but I've been struggling with this one for a bit and got nowhere.)

After reading Suresh's answer below, I can't help but think that there must be some colorful Helly theorem on manifolds, of which the question above is a special case. At the moment I don't even have a meaningful formulation of what this theorem could be, has it been treated before?

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# Covering a circle with red and blue arcs

We have a circle and two families of $n$ red arcs and $n$ blue arcs, positioned on the circle so that every two arcs of different colors intersect. Can one show that there is a point in the perimeter which is part of at least $n$ arcs?

(The statement sounds very simple. It makes me think the answer should be very simple too, but I've been struggling with this one for a bit and got nowhere.)