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$H=(V_1\cup V_2 \cup V_3, E)$ is a complete $3$ partite graph such that $|V_1|=|V_2|=|V_3|=n$ . Color the edges with three colors.

My question is: Is it possible to find sets $V_1' \subset V_1, V_2' \subset V_2$ and $V_3' \subset V_3$, such that all edges between $V_1', V_2'$ and $V_3'$ have the same color and also $|V_i'|\geq \epsilon n$ for $i=1,2,3.$?

Could someone please give a reference for this problem?

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  • $\begingroup$ By "color the triangles", you just mean color the edges, don't you? $\endgroup$ – LeechLattice Sep 17 '18 at 5:03
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    $\begingroup$ Assign a color to a triangle means to color the three edges of the triangle with that special color. $\endgroup$ – Joe Sep 17 '18 at 5:09
  • $\begingroup$ @bof: Thank you for your suggestion. I just edited. $\endgroup$ – Joe Sep 17 '18 at 17:08
  • $\begingroup$ This fails very hard, as you can colour all of the edges from $V_1$ to $V_2$ red and all other edges blue and not even have any monochromatic triangles. $\endgroup$ – Ben Barber Sep 18 '18 at 9:23
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Of course no, even for the bipartite graph and two colors. Considering the random coloring we get that the expectation of the number of such sets is exponentially small.

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  • $\begingroup$ Thanks for your response. What is the largest size that we could get? $\endgroup$ – Joe Sep 17 '18 at 17:58
  • $\begingroup$ $O(\log n)$ (the optimal constant is almost certainly unknown) $\endgroup$ – Fedor Petrov Sep 17 '18 at 18:45
  • $\begingroup$ Could you please provide me a reference? I would like to see the solution. Thank you! $\endgroup$ – Joe Sep 17 '18 at 19:12

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