three-uniform hypergraph on $n$ vertices with at least $n/3$ edges contains an independent set of size at least $\frac{2n^{3/2}}{3\sqrt{3m}}$

Question

Here is question 3 from chapter 3 Part 1 of The Probabilistic Method, 4th edition.

Prove that every three-uniform hypergraph with $n$ vertices and $m \ge n∕3$ edges contains an independent set (i.e., a set of vertices containing no edges) of size at least $\frac{2n^{3/2}}{3\sqrt{3m}}$.

I'm trying to solve this using a probabilistic approach. I've only been able to get $\frac{n^{3/2}-mn^{1/2}}{\sqrt{3m}}$ as a bound by defining $S\subset [n]$ by randomly including a vertex in $S$ with probability $p$. We call the resulting induced graph $G':=G[S]$ and denote its vertex set and edge set by $V'$ and $E'$. Then $\mathbb E[|V'|]-\mathbb E[|E'|]=np-mp^3$ because for each edge to be inside $S$ we need all three of its vertices to be chosen. Also, removing one vertex per edge in $G'$ results in an edge-less graph, so $\alpha(G)\ge\alpha(G')\ge np-mp^3$, where $\alpha$ is the size of the largest independent set. The inequality comes from taking the expectation on $\alpha(G')\ge \mathbb E[|V'|]-\mathbb E[|E'|]$.

Then the idea is to optimize over $p$, so make the 1st derivative w.r.t. $p$ equal to zero and get $p=\frac{n^{1/2}}{\sqrt{3m}}$ and the optimized RHS of the equality is $\frac{n^{3/2}-mn^{1/2}}{\sqrt{3m}}$.

Now the problem is that there is a minus there and the hypothesis is that $m\ge n/3$ so $-mn^{1/2}\le-\frac{-n^{3/2}}{3}$ and so $\alpha(G)\ge f(m,n)$ with $f(m,n)\le \frac{2n^{3/2}}{3\sqrt{3m}}=:B$, which does't prove anything. I think I must have made a mistake somewhere...

I only showed that the largest independent set is bounded from below by something which is below $B$. If I had showed that it is bounded by $B$ then that would probably be enough to say that there exists an independent set of size at least $B$.

Answer 1

Your algebra is fine. I think you just misunderstand what optimizing over $p$ does.

The inequality $$ \alpha(G) \ge np - mp^3 $$ holds for any $m, n$ and any $p \in [0,1]$, because the random process you've described works for any $m, n$ and any $p \in [0,1]$. It will produce a random independent set $S$, with $\mathbb E[|S|] \ge np - mp^3$; therefore, with positive probability, it will produce an $S$ with $|S| \ge np - mp^3$; therefore, the largest independent set has size at least $np - mp^3$.

At this point, if all you want to do is to write a proof, you can just write:

For no particular reason, we are inspired to take $p = \sqrt{\frac{n}{3m}}$, which satisfies $0 \le p \le 1$ provided that $m \ge n/3$. With this value of $p$, we conclude that $\alpha(G) \ge \frac{2n^{3/2}}{3\sqrt{3m}}$, which was what we wanted.

If we want to actually find a good value of $p$ to pick, so that we can come up with the proof, then optimizing over $p$ is a good idea. When $m \ge n/3$, we conclude that $p = \sqrt{\frac{n}{3m}}$ is the best probability to use. In other words, we deduce the inequality $$np - mp^3 \le \frac{2n^{3/2}}{3\sqrt{3m}}.$$ You're right that sticking this inequality into our proof would produce nonsense! But this inequality doesn't belong in our proof to begin with. It is part of our outside-the-proof reasoning for picking $p$.

All we actually need to know is that the right-hand side $\frac{2n^{3/2}}{3\sqrt{3m}}$ is achievable by some choice of $p$. We don't, strictly speaking, need to know that it's the largest possible value of $np - mp^3$. (But if it weren't, we could have written a better proof.)

In more complicated problems, it is common to skip taking the derivative, because we wouldn't get a closed form for the answer. Even here, we could be lazy and think "hmm... we want $np$ and $mp^3$ to be about the same size. Let's try setting $mp^3 = \frac12 np$. This gives us $p^2 = \frac{n}{2m}$, which gives some answer of the form $O(n^{3/2}/m^{1/2})$. Good enough!" Then, we could write a proof using $p = \sqrt{\frac{n}{2m}}$. That proof would work equally well, even though it no longer proves the best lower bound that this method can prove.