@ganeshie8

Question

@ganeshie8

parthkohli · Answer

Show that if the 21 edges of a complete 7-gon (all vertices joined to one another) is coloured red and blue, there exist at least three monochromatic triangles.

parthkohli · Answer

Obviously a pigeonhole thing, but how do I approach it?

ganeshie8 · Answer

|dw:1446569391614:dw|

parthkohli · Answer

Question: can an edge be a part of two triangles which satisfy this? I think yes.

ganeshie8 · Answer

Yes, each of the $21$ edges is formed by choosing $2$ vertices from the available $7$ vertices : $\binom{7}{2}$

parthkohli · Answer

Yes, of course. The simplest answer to my question is to just consider all edges coloured red.

ganeshie8 · Answer

I need to go for dinner, but here is what I have in mind : 

|dw:1446570150329:dw|

ganeshie8 · Answer

1)  From the vertex $A$, there are $6$ edges shooting out.

2) By pigeonhole principle at least $\dfrac{6}{2}=3$ of them will have same color (call it red for definiteness).

3) ...