Set Systems with Finite Chromatic Number

On the chromatic number of set systems

An r l -system is an r-uniform hypergraph in which every set of l vertices lies in at most one edge. Let mk r l be the minimum number of edges in an r l -system that is not k-colorable. Using probabilistic techniques, we prove that ar l kr−1 ln k l/ l−1 ≤ mk r l ≤ br l kr−1 ln k l/ l−1 where br l is explicitly defined and ar l is sufficiently small. We also give a different argument proving (fo...

On the chromatic number of graphs and set systems

On Chromatic Number of Graphs and Set-systems

Let a be a cardinal number . A graph V is said to have chromatic number a if x is the least cardinal such that, the set of vertices of ~ is the union of a sets, where no two elements of the same set are connected by an edge in ~§ . A graph 16~ is said to have colouring-number a, if a is the least cardinal such that the set of vertices of S has a well-ordering -< satisfying the condition that fo...

The set chromatic number of a graph

For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) 6= NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is calle...

The set chromatic number of random graphs

In this paper we study the set chromatic number of a random graph G(n, p) for a wide range of p = p(n). We show that the set chromatic number, as a function of p, forms an intriguing zigzag shape.