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Consider a set $A$ with $|A|=n \ge 3$, the set $A_2 = \{\{a,b\} : a,b \in A\}$, the set $A_3 = \{\{a,b,c\} : a,b,c \in A\}$, a function $f: A_2 \to \{0,1\}$ and the set $C_2 = \{\{a,b\} : {a,b \in A} \land {f(\{a,b\}) = 1}\}$.

Suppose we know that $|C_2| \ge q|A_2| = q{n \choose 2}$, $0 \lt q \le 1$.

Now let $C_3 = \{\{a,b,c\} : {a,b,c \in A} \land {f(\{a,b\}) = 1} \land {f(\{a,c\}) = 1} \land {f(\{b,c\}) = 1}\}$.

Is it always true that:

$$|C_3| \ge q^3|A_3| = q^3{n \choose 3}?$$

I am a little doubtful because $\{a,b\}$, $\{a,c\}$ and $\{b,c\}$ don't look fully independent because they share one element between themselves.

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  • $\begingroup$ Do you really mean $A_2 = \{ \{a,b\} \mid a,b \in A \}$, or do you mean that $A_2$ is the set of 2-element subsets of $A$ as you say in the title? These are not quite the same… $\endgroup$ Oct 3 at 15:54
  • $\begingroup$ @PeterLeFanuLumsdaine How are they different? $\endgroup$
    – Vincent
    Oct 3 at 16:08
  • $\begingroup$ @Vincent: $\{ \{a,b\} \mid a\in A\}$ doesn’t specify that $a$ and $b$ must be distinct, so it also includes singletons $\{x,x\} = \{x\}$. $\endgroup$ Oct 3 at 16:23
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    $\begingroup$ Well, I am not a mathematician, but I thought $a \not = b$ is implicit in the set notation $\{a,b\}$. Anyway this is what I mean. I might make it explicit if needed. $\endgroup$
    – BillyJoe
    Oct 3 at 18:08
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    $\begingroup$ @BillyJoe: In some contexts, yes, people write things like “Take some set $\{a,b\}$” and implicitly mean $a,b$ to be distinct. I think most would consider that a little sloppy, or informal at best. But distinctness is never be taken as implicit in a set-forming operation like $\{ \underline{\qquad} \mid a,b \in A \}$ — there, like in a quantification “for all $a, b \in A$”, it’s unambiguous that it ranges over all $a,b$ satisfying the constraints explicitly given. $\endgroup$ Oct 3 at 21:23

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To put it more visually: suppose you have a graph in which at least a fraction $q$ of pairs of vertices are connected by an edge, or equivalently: in which at least a fraction $q$ of possible edges actually exist. Your question is then if at least a fraction $q^3$ of possible triangles must exist as well.

Put like this it seems fairly obvious that the answer is no: it is easy to imagine a graph with a non-zero number of edges (and hence non-zero $q$) but no triangles at all.

(Caveat: this sort of assumes you are talking about all $n$ and all $q$ chosen in whatever order you like. My simple argument does not rule out a version of your claim starting with: 'For every $q \in (0, 1)$ there is an $N$ such that for all $n > N$...'. I'm not sure if some statement like that could be true or not.)

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