Is there a kind of combinatorial design that controls the sizes of small unions of the blocks?
I'm looking for a set $B$ of $|B|=b$ blocks, where $b\sim100$, which are subsets of a set $X$ of $|X|=v$ points, $v=32$ or less preferably, $v=64$. The blocks don't need to have a uniform size, but they should all be nonempty. More generally, given a union of a "small" set of blocks, it should be possible to infer the number of blocks in the union. To be precise, there should be an integer $\ell\geq 2$, ideally as large as possible, such that $B$ has the following property:
- If $P$ and $Q$ are subsets of $B$ with $|P|<|Q|\leq \ell$, then $\left|\bigcup P\right|<\left|\bigcup Q\right|$.
Note that if $b\leq v$ then this would be trivial. Just let $B$ be a set of $b$ distinct singleton subsets of $X$. Then for all $P\subset B$ we get $\left|\bigcup P\right|=|P|$, so the highlighted property is guaranteed for all $\ell$.
So the whole point of this question is asking if we can achieve some compression, i.e. $b>v$. Note that in the $b>v$ regime we can't possibly achieve $\ell=b$ anymore, but I'd still like for $\ell$ to be large.
For example, perhaps there is a design with $b=128$, $v=32$, and $\ell=3$, such that every block has size $4$, every union of two blocks has size $6$, $7$, or $8$, and every union of three blocks has size $9$ or more.
Arguments that limit $\ell$ or construct nice designs would be helpful. References to what these things are called would also be helpful!
Edit: According to https://www.win.tue.nl/~aeb/codes/Andw.html we have these nice constant-weight binary codes to work with:
- $A(64, 6, 4) = 336$ citing Brouwer et al (1990) "A new table of constant weight codes"
- $A(64, 8, 5) = 192$ citing a Steiner system $S(2,5,65)$. (How does that work?)
- $A(63, 10, 6) = 126 $ citing Smith et al (2006) "A New Table of Constant Weight Codes of Length Greater than 28" who list it on page 12, Table 7, bottom row as "CC" meaning a construction from a cyclic permutation
These constructions give us blocks that intersect each other at at most one point. From that, we can achieve $\ell=3$. In fact, the last construction of weight $6$ almost gets us to $\ell=4$:
- Each block has weight $6$.
- A union of two blocks has weight $11$ or $12$.
- A union of three blocks has weight $15$ to $18$.
- A union of four blocks has weight $18$ to $24$.
If there were some argument why a union of four blocks could be forced to have weight $\geq 19$, then we'd get $\ell=4$ after all.
Given those results, my questions are, more specifically:
- Are these sorts of codes the best bet for satisfying the inequality that I want?
- Is there a way to get to $\ell=4$ with $b\geq100$ and $v\leq64$?
- Is there a literature on this question that I've missed?