Intereting Posts

Let K be a field, and $I=(XY,(X-Y)Z)⊆K$. Prove that $√I=(XY,XZ,YZ)$.
Is the sum and difference of two irrationals always irrational?
Proof that there are infinitely many prime numbers starting with a given digit string
What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?
What is the categorical diagram for the tensor product?
Why square units?
Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems
If H ≤ Z(G) ≤ G, where G is a finite group,Z(G) is its center, and (G:H) = p for some prime p, then G is abelian.
Combinatorially prove that $\sum_{i=0}^n {n \choose i} 2^i = 3^n $
Is there a compactly supported smooth function which is exactly k times differentiable at exactly one point?
Rational Point in circle
shock waves characteristics
How to prove that $a^2b+b^2c+c^2a \leqslant 3$, where $a,b,c >0$, and $a^ab^bc^c=1$
Tensor product and injective maps
Simple proof of showing the Harmonic number $H_n = \Theta (\log n)$

Show ${1\over n}\sum|S_i|=O(\sqrt n)$ where $S_i\subset [n]$, $|S_i\cap S_i|\le 1$ for $i\ne j$. A previous question required showing $|E|\le {1\over 2}(\sqrt{t-1}n^{3\over 2}+n)$, for an $n$-vertex graph $G$. What I tried to do is placing $S_1,…,S_n$ in a row, and below it setting $1,…,n$ likewise. While I didn’t mind the fact that as a graph, $S_i$ cannot be connected to $S_j$ as it will have no meaning here, I did pay attention to the restrictions I am given: I can’t have a $K_{2,2}$ graph (also not a $K_{2,t\ge 2}$, but $t=2$ is tighter.). Is what I am doing admissible? If so, I am arriving at: $|E|\le {1\over 2}(\sqrt 2n^{3\over 2}+2n)$. Acknowledge(or at least assuming) that the number of edges is simply the sum of $|S_i|$’s, I get ${1\over n}\sum|S_i|\le \sqrt2n^{1\over 2}+2$ which seems $O(\sqrt n)$. Is it graph theory acceptable arguing? I would really appreciate your insights.

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If you assume to have $k$ almost-disjoint (that means $|S_i\cap S_j|\leq 1$ for any $i\neq j$) subsets of $\{1,2,\ldots,n\}$, any element $m\in\{1,2,\ldots, n\}$ can be labeled according to the subsets $S_i$ to which it belongs. Assume that $m$ belongs to $c(m)$ different subsets: then $\sum |S_i| = \sum_{m=1}^{n} c(m)$, but we must have

$$ \sum_{m=1}^{n}\binom{c(m)}{2}\leq \binom{k}{2}\leq\binom{n}{2} $$

so, by the Cauchy-Schwarz inequality, it follows that:

$$ \left(-n+\sum_{m=1}^n c(m)\right)^2 \leq n^3 $$

and:

$$\frac{1}{n}\sum |S_i| \leq \sqrt{n}+1.$$

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