Short proofs of three combinatorial results in the Johnson scheme

Authors: Danila Cherkashin, Yakov Shubin

math.CO2026-05v1license

Abstract

In this note, we give short proofs of three theorems concerning extremal problems in the Johnson scheme, or, in other terminology, on $(n,k,L)$ -systems. The main result is a proof of the Aljohani--Bamberg--Cameron conjecture which claims that if $n > n_0(k)$ and there are an $(n,k,L)$ -system and an $(n,k,\{0,\dots,k-1\}\setminus L)$ -system whose sizes have product $\binom{n}{k}$ , then they are a $t$ -intersecting family and a Steiner system $S(t,k,n)$ for some $t$ .

Cite

@article{arxiv.2605.30092,
  title  = {Short proofs of three combinatorial results in the Johnson scheme},
  author = {Danila Cherkashin and Yakov Shubin},
  journal= {arXiv preprint arXiv:2605.30092},
  year   = {2026}
}

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