## Generalized Kneser coloring theorems with combinatorial proofs (2001)

Venue: | INVENTIONES MATH |

Citations: | 20 - 3 self |

### BibTeX

@ARTICLE{Ziegler01generalizedkneser,

author = {Günter M. Ziegler},

title = {Generalized Kneser coloring theorems with combinatorial proofs},

journal = {INVENTIONES MATH},

year = {2001},

volume = {147},

pages = {2002}

}

### OpenURL

### Abstract

The Kneser conjecture (1955) was proved by Lovasz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matousek provided the rst combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, Alon-Frankl-Lovasz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.

### Citations

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(Show Context)
Citation Context ... and in which two k-subsets of an n-set are connected by an edge if they are disjoint, needs n 2k + 2 colors for a proper vertex coloring, if n 2k 4. Kneser's conjecture wassrst proved by Lovasz [Lo=-=v-=-78], in one of thesrst, and most spectacular, applications of an Algebraic Topology result (the Borsuk-Ulam theorem) to a combinatorial problem. An alternative proof was later given by Barany [Bar78],... |

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Citation Context ... signed color sets, so we obtain a simplicial map. This map is equivariant. Topologically, d 1 (n) is a simplicial (d 1)-sphere, namely the barycentric subdivision of the topological representation [=-=FL78-=-] of the alternating oriented matroid of rank d on n elements, see [BL78, Ex. 3.8] [BSZ + 99, Chap. 5/Sect. 9.4] [Zie93]. Similarly, d 2 (d) is a simplicial (d 2)-sphere. Both spheres have natural an... |

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Citation Context ... cone is dealt with by Lemma 6.1. 3 3 Colorings and Colorability Defects For s = (1; :::; 1), the following coloring of the Kneser hypergraphs is due to Kneser [Kne55] in the case r = 2 and to Erd}os [Erd76] in the general case. It corrects the coloring given in [Sar90, (3.3)]. Lemma 3.1. For r 2, k 2, constant s = (s; : : : ; s) with 1 ssr, and sn kr, KG r s [n] k 1 + 1 j r 1 s k ns rk+... |

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Citation Context ...f an Algebraic Topology result (the Borsuk-Ulam theorem) to a combinatorial problem. An alternative proof was later given by Barany [Bar78], extensions by Schrijver [Sch78], Alon, Frankl & Lovasz [AFL=-=8-=-6], Dol'nikov [Dol'88], Sarkaria [Sar90], andsnally by Kriz [Kri92, Kri00], whose result implies the theorems by Lovasz, Dol'nikov and Alon-Frankl-Lovasz. All of these were proved using Algebraic Topo... |

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Citation Context ...d 1)-sphere, namely the barycentric subdivision of the topological representation [FL78] of the alternating oriented matroid of rank d on n elements, see [BL78, Ex. 3.8] [BSZ + 99, Chap. 5/Sect. 9.4] =-=[Zie93-=-]. Similarly, d 2 (d) is a simplicial (d 2)-sphere. Both spheres have natural antipodal actions, and the map bc respects these. Thus, the Borsuk-Ulam theorem completes a topological proof at this poi... |

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Citation Context ...logical result used in this context, the Borsuk-Ulam theorem, has a variety of \combinatorial proofs," that is, reductions via simplicial approximation to combinatorial results such as Tucker's l=-=emma [T-=-uc46], the Ky Fan lemma [Fan52], etc. Supported by Deutsche Forschungs-Gemeinschaft (DFG), and by the Miller Institute at UC Berkeley 1 In 2000, Jir Matousek provided two breakthroughs for this situa... |

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Citation Context ..., then there are signed sets (A + ; A ) and (B + ; B ) such that (A + ; A ) = (B + ; B ), with A + B + and A B . This lemma has simple combinatorial proofs, e. g. by the method of Freund & Todd [FT8=-=1-=-]; see [Mat01]. (For further combinatorial Tucker lemmas, see Aigner [Aig01].) Theorem 4.2 (Dol'nikov [Dol'88]). For every hypergraph S 2 [n] , the 2-colorability defect is a lower bound for the chro... |

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14 |
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(Show Context)
Citation Context ...d generalization by (hyper)graph coloring theorems of Dol'nikov, Alon-Frankl-Lovasz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem. 1 Introduction Kneser's conjecture [=-=Kne5-=-5] stated that every coloring of the graph KG 2 [n] k , which has vertex set [n] k , and in which two k-subsets of an n-set are connected by an edge if they are disjoint, needs n 2k + 2 colors for a... |

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(Show Context)
Citation Context ...below be used for more general \signed sets," where elements get signs from Z p . The Borsuk-Ulam theorem asserts that there is no Z 2 -equivariant (continuous) map from S d to S d 1 . Dold's the=-=orem [Dol83]-=- is a transformation group extension of this: for every Z p -equivariant map f : X ! Y between free Z p -spaces (compact CW complexes, say) the dimension of Y is larger than the connectivity of X. The... |

11 |
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(Show Context)
Citation Context ...most spectacular, applications of an Algebraic Topology result (the Borsuk-Ulam theorem) to a combinatorial problem. An alternative proof was later given by Barany [Bar78], extensions by Schrijver [Sc=-=h-=-78], Alon, Frankl & Lovasz [AFL86], Dol'nikov [Dol'88], Sarkaria [Sar90], andsnally by Kriz [Kri92, Kri00], whose result implies the theorems by Lovasz, Dol'nikov and Alon-Frankl-Lovasz. All of these ... |

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(Show Context)
Citation Context ...ma [Fan52], etc. Supported by Deutsche Forschungs-Gemeinschaft (DFG), and by the Miller Institute at UC Berkeley 1 In 2000, Jir Matousek provided two breakthroughs for this situation. First, in [Mat0=-=1]-=- he provided a combinatorial bypass of the Borsuk-Ulam theorem, and thus a combinatorial proof of Kneser's conjecture. This used only an entirely combinatorial special case of the Tucker lemma. Second... |

6 |
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(Show Context)
Citation Context ... presented a lower bound for the chromatic numbers of hypergraphs KGr sS, “generalized r-uniform Kneser hypergraphs with intersection multiplicities s.” It generalized previous lower bounds by Kˇríˇz =-=[1, 2]-=- for the case s = (1, . . . , 1) without intersection multiplicities, and by Sarkaria [4] for S = ([n]) k . The following two problems that arise for intersection multiplicities si > 1 were noticed by... |

4 | A combinatorial inequality - Dol’nikov - 1988 |

3 | On the chromatic number of Kneser hypergraphs
- Matousek
- 2000
(Show Context)
Citation Context ...vided a combinatorial bypass of the Borsuk-Ulam theorem, and thus a combinatorial proof of Kneser's conjecture. This used only an entirely combinatorial special case of the Tucker lemma. Secondly, in =-=[Mat-=-00], Matousek gave a simple and elegant derivation of Kriz' theorem from Dold's theorem. Here, we shall demonstrate the power and extend the scope of Matousek's approach, by establishing a simple com... |

2 |
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- 2001
(Show Context)
Citation Context ... ) = (B + ; B ), with A + B + and A B . This lemma has simple combinatorial proofs, e. g. by the method of Freund & Todd [FT81]; see [Mat01]. (For further combinatorial Tucker lemmas, see Aigner [Ai=-=g-=-01].) Theorem 4.2 (Dol'nikov [Dol'88]). For every hypergraph S 2 [n] , the 2-colorability defect is a lower bound for the chromatic number, (KG 2 S) cd 2 S: Combinatorial Proof. Let c : S ! [m] be a... |

2 | On generalized Kneser hypergraph colorings
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- 2005
(Show Context)
Citation Context ...resence of intersection multiplicities, there are two different versions of a “Kneser hypergraph,” depending on whether one admits hypergraph edges that are multisets rather than sets. It is shown in =-=[3]-=- that the chromatic numbers are substantially different for the two concepts of hypergraphs. The lower bounds of Sarkaria [4] and of [5, Thm. 5.1] apply only to the multiset version. The coloring and ... |