TY - JOUR

T1 - Hypergraph list coloring and Euclidean Ramsey theory

AU - Alon, Noga

AU - Kostochka, Alexandr

PY - 2011/10

Y1 - 2011/10

N2 - A hypergraph is simple if it has no two edges sharing more than a single vertex. It is s-list colorable (or s-choosable) if for any assignment of a list of s colors to each of its vertices, there is a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic. We prove that for every positive integer r, there is a function dr(s) such that no r-uniform simple hypergraph with average degree at least dr(s) is s-list-colorable. This extends a similar result for graphs, due to the first author, but does not give as good estimates of dr(s) as are known for d2(s), since our proof only shows that for each fixed r ≥ 2, dr(s) ≤ 2crsr-1. We use the result to prove that for any finite set of points X in the plane, and for any finite integer s, one can assign a list of s distinct colors to each point of the plane so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of X.

AB - A hypergraph is simple if it has no two edges sharing more than a single vertex. It is s-list colorable (or s-choosable) if for any assignment of a list of s colors to each of its vertices, there is a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic. We prove that for every positive integer r, there is a function dr(s) such that no r-uniform simple hypergraph with average degree at least dr(s) is s-list-colorable. This extends a similar result for graphs, due to the first author, but does not give as good estimates of dr(s) as are known for d2(s), since our proof only shows that for each fixed r ≥ 2, dr(s) ≤ 2crsr-1. We use the result to prove that for any finite set of points X in the plane, and for any finite integer s, one can assign a list of s distinct colors to each point of the plane so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of X.

KW - Average degree

KW - Euclidean Ramsey Theory

KW - Hypergraphs

KW - List coloring

UR - http://www.scopus.com/inward/record.url?scp=80051872270&partnerID=8YFLogxK

U2 - 10.1002/rsa.20361

DO - 10.1002/rsa.20361

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:80051872270

SN - 1042-9832

VL - 39

SP - 377

EP - 390

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 3

ER -