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arXiv · 2605.30092

Short proofs of three combinatorial results in the Johnson scheme

Danila Cherkashin·Yakov Shubin
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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 $t$.

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Danila Cherkashin, Yakov Shubin. 2026-05-28. Short proofs of three combinatorial results in the Johnson scheme. https://arxiv.org/abs/2605.30092

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