556 research outputs found
Uniform hypergraphs containing no grids
A hypergraph is called an rĂr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that AiâŠAj=BiâŠBj=Ď for 1â¤i<jâ¤r and {pipe}AiâŠBj{pipe}=1 for 1â¤i, jâ¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1âŠC2{pipe}={pipe}C2âŠC3{pipe}={pipe}C3âŠC1{pipe}=1, C1âŠC2â C1âŠC3. A hypergraph is linear, if {pipe}EâŠF{pipe}â¤1 holds for every pair of edges Eâ F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For râĽ. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Š 2013 Elsevier Ltd
Coloring d-Embeddable k-Uniform Hypergraphs
This paper extends the scenario of the Four Color Theorem in the following
way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly)
embedded into R^d. We investigate lower and upper bounds on the maximum (weak
and strong) chromatic number of hypergraphs in H(d,k). For example, we can
prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak
chromatic number is Omega(log n/log log n), whereas the weak chromatic number
for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.Comment: 18 page
- âŚ