Advancing the R\"{o}dl Nibble: New bounds on matchings and the list chromatic index of hypergraphs

Abstract

Let $H$ be a $(k+1)$-uniform hypergraph which is nearly $D$-regular, such that any set of $i$ vertices is contained in at most $D_i$ edges of $H$ for each $i = 2, 3, \dots, k+1$. Influential results of Pippenger and of Frankl and R\"odl show that the \textit{R\"odl Nibble} -- a probabilistic procedure which iteratively constructs a matching in small bits -- can produce an almost-perfect matching in $H$, provided $D_2$ is much smaller than $D$. The quantitative aspects of this result were sharpened by several authors, with the previously best-known result due to Vu, whose result takes more of the codegree sequence $D_2, \dots, D_{k+1}$ into account. We improve Vu's result, by showing the R\"odl Nibble can ``exhaust'' the full codegree sequence up to one of several natural bottlenecks, even tolerating extensive ``clustering'' of codegree values. Up to a subpolynomial error term, we believe our result to be the optimal usage of pure nibble methodology. We also show that our matching can be taken to be ``pseudorandom'' with respect to a set of weight functions on $V(H)$, and we use this result to derive other hypergraph matching results in partite settings, including a new bound on the list chromatic index which implies the best-known result of Molloy and Reed up to the error term, and is stronger when the hypergraph is not close to linear, i.e.\ $D_2=\omega(1)$. We also apply our results to obtain improved bounds on almost-spanning structures in Latin squares and designs, and the maximum diameter of a simplicial complex.