3 corrected some typos

I like the question. You are actually asking, what is the smallest finite probability space (in size of $\Omega'$) on which one can have $d$ distinct pair-wisely independent (but not necessarily independent) events of probability $\frac{1}{2}$. Just to explain the reformulation: once the $i$-th event is defined to be $$E_{i}=\lbrace\omega\in\Omega'\;|\;\omega(i)=1\rbrace,$$ the "efficient substitute" criterion ammounts amounts to $\mathrm{Pr}(E_{i})=\frac{1}{2}$ and $\mathrm{Pr}(E_{i}\cap E_{j})=\frac{1}{4}$ for every $i,j$ with $1\le i,j\le d$ and $i\not=j$. The example for $d=3$ is a well known case of this situation.

Now, every such space satisfies $|\Omega'|=4n$ for some natural $n$ (the reason being obvious). Knowing this, we could ask an inverse question: given a $4n$-set $\Omega'$, what is the maximum size of a family $\mathcal{F}$ of $2n$-subsets of $\Omega'$ such that every two of them have intersection of size $n$. Knowing a precise answer to this, the original problem is solved as well: given a $d$ take the smallest $4n$ such that the maximum size of $\mathcal{F}$ is at lesat least $d$.

There is a paper titled "Pairwise intersections and forbidden configurations". Let me quote from the abstract:

Let $f_{m}(a,b,c,d)$ denote the maximum size of the family $\mathcal{F}$ of subsets of an $m$-element set for which there is no pair of subsets $A,B\in\mathcal{F}$ with $|A\cap B|\ge a$, $|A^{c}\cap B|\ge b$, $|A\cap B^{c}|\ge c$, and $|A^{c}\cap B^{c}|\ge d$.

The count we are looking for is exactly $f_{4n}(n+1,n+1,n+1,n+1)$. Besides other interesting things, the paper gives also asymptotic estimates for this count; one of them gives $\Theta(4n^{2n-1})$ in our case.

Edit: unfortunately, I was too quick with the asymptotic estimate. The paper gives an estimate only for fixed $a,b,c,d$ as $m$ tends to $\infty$.

2 correction

I like the question. You are actually asking, what is the smallest finite probability space (in size of $\Omega'$) on which one can have $d$ distinct pair-wisely independent (but not necessarily independent) events of probability $\frac{1}{2}$. Just to explain the reformulation: once the $i$-th event is defined to be $$E_{i}=\lbrace\omega\in\Omega'\;|\;\omega(i)=1\rbrace,$$ the "efficient substitute" criterion ammounts to $\mathrm{Pr}(E_{i})=\frac{1}{2}$ and $\mathrm{Pr}(E_{i}\cap E_{j})=\frac{1}{4}$ for every $i,j$ with $1\le i,j\le d$ and $i\not=j$. The example for $d=3$ is a well known case of this situation.

Now, every such space satisfies $|\Omega'|=4n$ for some natural $n$ (the reason being obvious). Knowing this, we could ask an inverse question: given a $4n$-set $\Omega'$, what is the maximum size of a family $\mathcal{F}$ of $2n$-subsets of $\Omega'$ such that every two of them have intersection of size $n$. Knowing a precise answer to this, the original problem is solved as well: given a $d$ take the smallest $4n$ such that the maximum size of $\mathcal{F}$ is at lesat $d$.

There is a paper titled "Pairwise intersections and forbidden configurations". Let me quote from the abstract:

Let $f_{m}(a,b,c,d)$ denote the maximum size of the family $\mathcal{F}$ of subsets of an $m$-element set for which there is no pair of subsets $A,B\in\mathcal{F}$ with $|A\cap B|\ge a$, $|A^{c}\cap B|\ge b$, $|A\cap B^{c}|\ge c$, and $|A^{c}\cap B^{c}|\ge d$.

The count we are looking for is exactly $f_{4n}(n+1,n+1,n+1,n+1)$. Besides other interesting things, the paper gives also asymptotic estimates for this count; one of them gives $\Theta(4n^{2n-1})$ in our case.

Edit: unfortunately, I was too quick with the asymptotic estimate. The paper gives an estimate only for fixed $a,b,c,d$ as $m$ tends to $\infty$.

1

I like the question. You are actually asking, what is the smallest finite probability space (in size of $\Omega'$) on which one can have $d$ distinct pair-wisely independent (but not necessarily independent) events of probability $\frac{1}{2}$. Just to explain the reformulation: once the $i$-th event is defined to be $$E_{i}=\lbrace\omega\in\Omega'\;|\;\omega(i)=1\rbrace,$$ the "efficient substitute" criterion ammounts to $\mathrm{Pr}(E_{i})=\frac{1}{2}$ and $\mathrm{Pr}(E_{i}\cap E_{j})=\frac{1}{4}$ for every $i,j$ with $1\le i,j\le d$ and $i\not=j$. The example for $d=3$ is a well known case of this situation.

Now, every such space satisfies $|\Omega'|=4n$ for some natural $n$ (the reason being obvious). Knowing this, we could ask an inverse question: given a $4n$-set $\Omega'$, what is the maximum size of a family $\mathcal{F}$ of $2n$-subsets of $\Omega'$ such that every two of them have intersection of size $n$. Knowing a precise answer to this, the original problem is solved as well: given a $d$ take the smallest $4n$ such that the maximum size of $\mathcal{F}$ is at lesat $d$.

There is a paper titled "Pairwise intersections and forbidden configurations". Let me quote from the abstract:

Let $f_{m}(a,b,c,d)$ denote the maximum size of the family $\mathcal{F}$ of subsets of an $m$-element set for which there is no pair of subsets $A,B\in\mathcal{F}$ with $|A\cap B|\ge a$, $|A^{c}\cap B|\ge b$, $|A\cap B^{c}|\ge c$, and $|A^{c}\cap B^{c}|\ge d$.

The count we are looking for is exactly $f_{4n}(n+1,n+1,n+1,n+1)$. Besides other interesting things, the paper gives also asymptotic estimates for this count; one of them gives $\Theta(4n^{2n-1})$ in our case.