why are you posted this problème 2 time :

Use Pigeon hole principle.(For more advanced cases refer to Ramsey numbers)

The Captain wrote: Use Pigeon hole principle.(For more advanced cases refer to Ramsey numbers) Can you explain it a little longer, please. I tried but got stuck

Choose a particular person of the group. He/She corresponds with$16$ others. By the Pigeonhole Principle, he/she must write to at least $6$of the people of one topic, say topic $I$. If any pair of these$6$ people corresponds on topic $I$,then he/she and this pair do the trick. Otherwise, these $6$ correspond amongst themselves only on topics$II or III$. Choose a particular person from this $six-group$.There must be $3$ of the $5$ remaining that correspond with that in one of the topics, say topic $II$. If amongst these three there is a pair that corresponds with each other on topic $II$, then he/she and this pair correspond on topic $II$. Otherwise, these three people only correspond with one another on topic $III$, and we are done.

The Captain wrote: Choose a particular person of the group. He/She corresponds with$16$ others. By the Pigeonhole Principle, he/she must write to at least $6$of the people of one topic, say topic $I$. If any pair of these$6$ people corresponds on topic $I$,then he/she and this pair do the trick. Otherwise, these $6$ correspond amongst themselves only on topics$II or III$. Choose a particular person from this $six-group$.There must be $3$ of the $5$ remaining that correspond with that in one of the topics, say topic $II$. If amongst these three there is a pair that corresponds with each other on topic $II$, then he/she and this pair correspond on topic $II$. Otherwise, these three people only correspond with one another on topic $III$, and we are done. Yes, it's by far trivial, but I wondered about the advanced cases you mentioned that used Ramsey Numbers. Anyway, thanks for your help.

Label the people $v_1, \ldots, v_{17}$. If two people discussed topic 1 connect them with red, if topic 2 connect with blue, if topic 3 connect with green. Note $v_1$ has degree 16, so by pigeonhole principle at least 6 of them is one color WLOG let it be red. If any of the six vertices $v_i, v_j$ are connected with red we have a monochromatic triangle $v_1,v_i,v_j$. If none of them are connected to each other with red, then they must be connected with blue or green. We can repeat the argument to find that there must be a blue monochromatic triangle or a green monochromatic triangle.