Intersection patterns of finite sets and of convex sets

The main result is a common generalization of results on lower bounds for the chromatic number of r-uniform hypergraphs and some of the major theorems in Tverberg-type theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser's conjecture. Moreover, combining this result with recent work of Mabillard and Wagner we show that the existence of certain equivariant maps yields lower bounds for chromatic numbers. We obtain an essentially elementary proof of the result of Schrijver on the chromatic number of stable Kneser graphs. In fact, we show that every neighborly even-dimensional polytope yields a small induced subgraph of the Kneser graph of the same chromatic number. We furthermore use this geometric viewpoint to give tight lower bounds for the chromatic number of certain small subhypergraphs of Kneser hypergraphs.

[1]  V. Dol'nikov,et al.  A certain combinatorial inequality , 1988 .

[2]  David Gale,et al.  Neighborly and cyclic polytopes , 1963 .

[3]  K. S. Sarkaria.  A generalized van Kampen-Flores theorem , 1991 .

[4]  Joshua Evan Greene,et al.  A New Short Proof of Kneser's Conjecture , 2002, Am. Math. Mon..

[5]  Uli Wagner,et al.  Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems , 2015, ArXiv.

[6]  Imre Bárány,et al.  On a Topological Generalization of a Theorem of Tverberg , 1981 .

[7]  P. Os,et al.  Problems and Results in Combinatorial Analysis , 1978 .

[8]  A. Volovikov,et al.  On the van Kampen-Flores theorem , 1996 .

[9]  Michal Adamaszek,et al.  Nerve Complexes of Circular Arcs , 2014, Discret. Comput. Geom..

[10]  I. Bárány,et al.  On a common generalization of Borsuk's and Radon's theorem , 1979 .

[11]  Florian Frick,et al.  Counterexamples to the topological Tverberg conjecture , 2015 .

[12]  Micha A. Perles,et al.  Strong General Position , 2014 .

[13]  Imre Bárány,et al.  A Short Proof of Kneser's Conjecture , 1978, J. Comb. Theory, Ser. A.

[14]  Murad Ozaydin,et al.  Equivariant Maps for the Symmetric Group , 1987 .

[15]  Günter M. Ziegler,et al.  Generalized Kneser coloring theorems with combinatorial proofs , 2002 .

[16]  Uli Wagner,et al.  Eliminating Higher-Multiplicity Intersections, III. Codimension 2 , 2015, Israel Journal of Mathematics.

[17]  Sinisa T. Vrecica,et al.  The Colored Tverberg's Problem and Complexes of Injective Functions , 1992, J. Comb. Theory, Ser. A.

[18]  J. Radon.  Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , 1921 .

[19]  J. Matousek,et al.  Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry , 2007 .

[20]  Noga Alon,et al.  The chromatic number of kneser hypergraphs , 1986 .

[21]  A. Schrijver,et al.  Vertex-critical subgraphs of Kneser-graphs , 1978 .

[22]  H. Tverberg.  A Generalization of Radon's Theorem , 1966 .

[23]  J. Matousek,et al.  Using The Borsuk-Ulam Theorem , 2007 .

[24]  STABLE KNESER HYPERGRAPHS AND IDEALS IN N WITH THE NIKODÝM PROPERTY , 2008 .

[25]  Wolfgang Kühnel.  Higherdimensional Analogues of Csaszar's Torus , 1986 .

[26]  Günter M. Ziegler,et al.  On generalized Kneser hypergraph colorings , 2007, J. Comb. Theory, Ser. A.

[27]  Edward D. Kim,et al.  Jahresbericht der deutschen Mathematiker-Vereinigung , 1902 .

[28]  Hossein Hajiabolhassan,et al.  On the chromatic number of general Kneser hypergraphs , 2013, J. Comb. Theory B.

[29]  N. Alon,et al.  Stable Kneser hypergraphs and ideals in $\mathbb {N}$ with the Nikodym property , 2008 .

[30]  M. Gromov.  Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry , 2010 .

[31]  Igor Kriz.  A correction to “Equivariant cohomology and lower bounds for chromatic numbers” , 1999 .

[32]  László Lovász,et al.  Kneser's Conjecture, Chromatic Number, and Homotopy , 1978, J. Comb. Theory A.

[33]  Karanbir S. Sarkaria,et al.  A generalized kneser conjecture , 1990, J. Comb. Theory, Ser. B.

[34]  Pavle V. M. Blagojevi'c,et al.  Optimal bounds for the colored Tverberg problem , 2009, 0910.4987.

[35]  H. Thomas,et al.  Higher-dimensional cluster combinatorics and representation theory , 2010, 1001.5437.

[36]  Florian Frick,et al.  Barycenters of polytope skeleta and counterexamples to the Topological Tverberg Conjecture, via constraints , 2015, Journal of the European Mathematical Society.

[37]  G. Ziegler,et al.  Generalized Kneser coloring theorems with combinatorial proofs , 2001, math/0103146.

[38]  Arnau Padrol,et al.  Many Neighborly Polytopes and Oriented Matroids , 2012, Discret. Comput. Geom..

[39]  Florian Frick,et al.  Tverberg plus constraints , 2014, 1401.0690.

[40]  I. Kríz.  A correction to “Equivariant cohomology and lower bounds for chromatic numbers” , 1992 .

[41]  Eth Zentrum,et al.  A Combinatorial Proof of Kneser's Conjecture , 2022 .

[42]  P. Erdos.  Problems and Results in Combinatorial Analysis , 2022 .

[43]  Frédéric Meunier,et al.  The chromatic number of almost stable Kneser hypergraphs , 2009, J. Comb. Theory, Ser. A.

[44]  Uli Wagner,et al.  Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range , 2016, SoCG.