Six points connected in pairs by coloured lines Six points are connected in pairs by lines each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. A path is closed if it begins and ends at the same point.

Assume that the red is the most represented color. So there are at least 8 red lines. If there is not a blue 4-cycle among $A,B,C,D$, there must be at least 2 red lines among $A,B,C,D$ sharing a common vertex, say $AB$ and $AC$. Between $\{E,F\}$ and $\{A,B,C,D\}$ there are 8 lines: at least $5$ of them must be red.
Assume that $EF$ is red. If all the $4$ lines between $E$ (or $F$) and $\{A,B,C,D\}$ are red, $ACEB$ ($ACFB$) is a red 4-cycle. So we can assume that there are $3$ red lines between $E$ and $\{A,B,C,D\}$ (and we can assume that they are $EA,EC,ED$) and $2$ red lines between $F$ and $\{A,B,C,D\}$, so at least $1$ red line between $F$ and $\{A,B,C\}$, completing one of the following $4$-cycles: $ABFE$,$AFEC$,$ACFE$. If $EF$ is blue, then there are exactly $3$ red lines between $F$ and $\{A,B,C,D\}$: $FA,FB,FD$, completing the $4$-cycle $AFDE$.
This proves that there is a monochromatic $C_4$ in any $2$-coloring of the edges of $K_6$.
