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I was given the next question: Let $Fn$ be the graph that's defined this way:

$Vn$ (vertices) equals to the set of all subsets of {1,2,...,n} such that their cardinality is exactly 2

$En$ (edges) equals to the set of all subsets of $Vn$ such that their cardinality is exactly 2 and {A.B} belongs to $En$ if and only if A and B have no elements in common.

I was asked to find the smallest $n$ (a natural number) such that for every $m$ that is bigger than/equals to $n$, $Fn$ is connected. I tried drawing some examples and I feel as if I'm stuck..any clues or hints will be very helpful! Thank you so much!!

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    If $m\geq 6$, then $F_m$ is obviously connected. Can you see why ? If you see this, the n you only have to check whether $F_5, F_4, F_3$ are connected ($F_1$ has no vertex as there is no subset of $\{1\}$ of cardinality $2$, $F_2$ has only one vertex so it's connected) – 2017-02-06
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    But what promises me that m>=6 is connected? I can see that for m=6 it works, m=7 it does work but what about the rest? induction? – 2017-02-06
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    Let $A,B$ be vertices of $F_m$ with $m\geq 6$. Then $A\cup B$ has at most $4$ elements, and since we have $m\geq 6$, there are at least two elements $c,d\notin A\cup B$, so that $T:=\{c,d\}$ is an vertex in $F_m$, and has no elements in common with $A$, or $B$, so that both $\{A,T\}$ and $\{B,T\}$ are edges in $F_m$, which shows that there is a path from $A$ to $B$ – 2017-02-06
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    Thanks Max! I have one question though, is there a way to look at these questions? How did you figure out that it's the right way? I find myself struggling with these types of questions.. – 2017-02-06
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    Mhm, well the way I found it was simply thinking "What could make $A$ and $B$ be connected ?" and I tried seeing if by any chance there could be a path of length 2 between them (there usually is, in these kinds of exercises), and then 6 was obvious. I think the best thing I could say is that you need practice to solve those little exercises. If you practice enough, struggling hard enough, then at some point you should begin noticing those things – 2017-02-06

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