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Draw the complete graph $K_n$ in the plane [added: in general position] with every edge a straight line and randomly label the edges $0$ or $1$. Does this graph always have a spanning tree with no edges crossing and edge-labels either all $0$ or all $1$?

Draw the complete graph $K_n$ on a plane in general position with every edge a straight line and randomly label the edges $0$ or $1$. Does this graph always have a spanning tree with no edges crossing and edge-labels either all $0$ or all $1$?

Draw the complete graph $K_n$ in the plane with every edge a straight line and randomly label the edges $0$ or $1$. Does this graph always have a spanning tree with no edges crossing and edge-labels either all $0$ or all $1$?

Draw the complete graph $K_n$ in the plane [added: in general position] with every edge a straight line and randomly label the edges $0$ or $1$. Does this graph always have a spanning tree with no edges crossing and edge-labels either all $0$ or all $1$?