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made it more concis, corrected some mistakes
myro
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The following construction shows that $c_3(n)\ge n^2/4-1$. Put $k=\lceil \frac{n+1}2\rceil$ and draw a regular $k$-gon $K$ with side $1$. Initially, we place $k$ black unit circles centered in the corners of $K$. This generates $k - 1$ circular layers of $k$ double intersection points (see, for instance, the last picture at the question). Since $k-1\ge n-k$, we can cover $n-k$ layers by red circles, achieving at least $k$ triple intersection points on each layer and $k(n-k)\ge n^2/4-1$ of them in total.

myro
  • 63
  • 7