a) Consider a regular hexagon with center point 7. WLOG assume point 7 is Red (R). If all of points 1,2,3,4,5,6 are Blue, then triangles 2,6,4 and 1,3,5 are Blue equilateral triangles — contradiction. So at least one vertex, say point 1, is Red. Then points 2 and 6 must both be Blue (otherwise triangle 1,6,7 or 1,2,7 is Red). Then point 4 must be Red (otherwise 2,6,4 is Blue). So points 1,7,4 form a Red equilateral triangle.
b) Points 3 and 5 must be Blue (otherwise 7,3,4 or 7,5,4 is Red). Now draw point 8 such that 8,6,5 form an equilateral triangle. If point 8 is Blue, then 5,6,8 is a Blue equilateral triangle. If point 8 is Red, then 1,4,8 is a Red equilateral triangle.