Abstract:Two subsets $A$ and $B$ of a ground set $X$ are crossing if none of the four sets $A\setminus B,B\setminus A,A\cap B, X\setminus (A\cup B)$ are empty. Almost fifty years ago, Karzanov and Lomonosov conjectured that every family of subsets of an $n$-element ground set with no $k$-pairwise crossing members has size $O(kn)$. We prove the bound $O_k(n)$, settling (arguably) the main problem about the growth rate of such families.
In fact, we prove the stronger statement that every $k$-wise laminar family has at most linear size. Here a family is $k$-wise laminar if it contains no antichain of size $k$ whose members have nonempty common intersection.
From: Istvan Tomon [view email]
[v1]
Thu, 5 Mar 2026 18:05:05 UTC (13 KB)
[v2]
Fri, 28 Aug 2026 13:58:26 UTC (14 KB)
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