The chromatic numbers of random hypergraphs

Abstract

For a pair of integers 1 F ␥ -r, the ␥-chromatic number of an r-uniform Ž . hypergraph H s V, E is the minimal k, for which there exists a partition of V into subsets < < T, . . . , T such that e l T F ␥ for every e g E. In this paper we determine the asymptotic

On $r$-colorability of random hypergraphs

“…Let us make intermediate conclusions. Our new lower bound (7) for the threshold probability of r-colorability of random hypergraph H(n, k, p) improves all previously known results in the following wide area (see condition (9) of Corollary 1 and condition ( 16) of Corollary 2):…”

On the chromatic number of a random hypergraph

“…Finally, we note that Krivelevich and Sudakov [14] studied a wide range of random hypergraph colouring problems, and some of their results were recently improved by Kupavskii and Shabanov [15]. But, in the setting of this paper, these results are much less precise than those we establish here.…”

On the 2-Colorability of Random Hypergraphs

“…where the last inequality holds for sufficiently large n. Combining (16) and (15) gives S < (W Y −(d−1)/2 + 1) g n max ≡ D × g n max which completes the proof. (We note that the constant D can be optimized by replacing our sums by integrals and using Laplace's method [2,9].…”