Ziegler–Alon–Drewnowski–Łuczak conjecture on stable Kneser hypergraphs

Let kk, rr, and nn be positive integers with nrkn\geq rk and r2r\geq2. An rr-stable kk-element subset of {1,,n}\{1,\dots,n\} is one in which any two elements have cyclic distance at least rr. The hypergraph KGr(k,n)r-stab\mathrm{KG}^r(k,n)_{r\text{-stab}} is the subhypergraph of the rr-uniform Kneser hypergraph induced by these rr-stable subsets.

Ziegler–Alon–Drewnowski–Łuczak conjecture.

χ(KGr(k,n)r-stab)=nr(k1)r1.\chi\bigl(\mathrm{KG}^r(k,n)_{r\text{-stab}}\bigr)=\left\lceil\frac{n-r(k-1)}{r-1}\right\rceil.

The source attributes this conjecture to Ziegler, Alon, Drewnowski, and Łuczak. The supplied text does not give evidence that it has been resolved.

Sources & referencesView supporting material

Primary source

Florian Frick, “Intersection patterns of finite sets and of convex sets”, arXiv:1607.01003 (2016).

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