This has a trivial counterexample for c > 0.
Consider X = {(0,0), (0,1), (1,0)}, colored {red, blue, blue} respectively. There is only one empty triangle in X, and it is not monochromatic. So it has 0 monochromatic empty triangles, and 0 is not > c*(3^2) for c > 0.
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This has a trivial
This has a trivial counterexample for c > 0.
Consider X = {(0,0), (0,1), (1,0)}, colored {red, blue, blue} respectively. There is only one empty triangle in X, and it is not monochromatic. So it has 0 monochromatic empty triangles, and 0 is not > c*(3^2) for c > 0.