Abstract. The chromatic number of a subset of the real plane is the smallest number of colors assigned to the elements of that set such that no two points at distance 1 receive the same color. It is known that the chromatic number of the plane is between 4 and 7. In this note, we determine the bounds on the chromatic number for several classes of subsets of the plane such as extensions of the rational plane, sets in convex position, infinite strips, and parallel lines.
In this paper, we address a particular variation of the Turán problem for the hypercube. Alon, Krech and Szabó (2007) asked "In an n-dimensional hypercube, Q n , and for ℓ < d < n, what is the size of a smallest set, S, of Q ℓ 's so that every Q d contains at least one member of S?" Likewise, they asked a similar Ramsey type question: "What is the largest number of colors that we can use to color the copies of Q ℓ in Q n such that each Q d contains a Q ℓ of each color?" We give upper and lower bounds for each of these questions and provide constructions of the set S above for some specific cases.
Abstract. For any positive integer r, an r-identifying code on a graph G is a set C ⊂ V (G) such that for every vertex in V (G), the intersection of the radius-r closed neighborhood with C is nonempty and pairwise distinct. For a finite graph, the density of a code is |C|/|V (G)|, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We find a code of density less than 5/(6r), which is sparser than the prior best construction which has density approximately 8/(9r).
An r-identifying code on a graph G is a set C ⊂ V (G) such that for every vertex in V (G), the intersection of the radius-r closed neighborhood with C is nonempty and unique. On a finite graph, the density of a code is |C|/|V (G)|, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of r in both the square and hexagonal grids.
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