Maximal number of $k$-subsets of an $n$-set such that any two subsets meet in at most $(k-2)$ points

Question

I am interested in the maximal number of $k$-subsets of an $n$-set such that any two subsets meet in at most $(k-2)$ points. I found that for $k=3$ and $k=4$, we have the sequences http://oeis.org/A001839 and http://oeis.org/A001843 as solutions which gives the impression that higher orders should be obtainable via $A(n,4,k)$ but this seems to be complicated when I look at the research invested in $A$.

"A binary constant weight code of word length $n$ and weight $w$ and minimum distance $d$ is a collection of $(0,1)$-vectors of length $n$, all having $w$ ones and $n - w$ zeros, such that any two of these vectors differ in at least $d$ places. The maximum size of such a code is denoted by $A(n; d;w)$." The definition of $A$ can be found here https://www.win.tue.nl/~aeb/preprints/cw4p.pdf.

I tried to say something about cases $k>5$ using the Erdös-Ko-Rado Theorem but I was not really succesfull. Is my intuition right, that $A(n,4,k)$ described exactly the quantity, which I am looking for?

Answer 1

You are looking for constant weight codes with maximum size given by $A(n,4,k).$ Yes this is a very hard problem in terms of constructions in general.

There is the following paper available online here which has some results:

Some new distance-4 constant weight codes A. E. Brouwer & T. Etzion 2010-02-18

Abstract

Improved binary constant weight codes with minimum distance 4 and length at most 28 are constructed. A table with bounds on the chromatic number of small Johnson graphs is given.

The paper has been published see here and also cited 10 times according to that link. I suggest you check out those links to see if improved bounds or constructions are available.