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

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Let kk, rr, and nn be positive integers with n≥rkn\geq rk and r≥2r\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)=⌈n−r(k−1)r−1⌉.\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.

References

Primary source

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

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