I’m reading about the Ramsey number, trying to understand the demonstrations of the exact values, in this case my question is about proof of $R (3,6)\leq 18$.
In this proof it is claimed that: all 5-cycles do not contain neither a triangle nor an independent set of three vertices, but it does not show the proof, it only gives a reference to Gleason’s “Combinatorial Relations and Chromatic Graphs”, in which there is no proof of this either.
How can I prove that this assertion: “all 5-cycles do not contain neither a triangle nor an independent set of three vertices”?
But this is obvious, 5-cycles does not contain triangles as it is a cycle of length 5. It does not contain an independent set of size 3 because it's maximum independent set can have size at most 2. This is because choose any vertex in the cycle to be in the independent set, then it's 2 neighbors can't be in the idependent set. You are left with 2 more connected vertices to be in the independent set. You can only choose one out of the 2. So you can have at most 2 vertices in the independent set. Is this your question?