Graphs of low average degree without independent transversals

An independent transversal of a graph G with a vertex partition P is an independent set of G intersecting each block of P in a single vertex. Wanless and Wood proved that if each block of P has size at least t and the average degree of vertices in each block is at most t/4, then an independent transversal of P exists. We present a construction showing that this result is optimal: for any ε>0 and sufficiently large t, there is a family of forests with vertex partitions whose block size is at least t, average degree of vertices in each block is at most (1/4+ε)t, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanless-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version

Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lov\'asz Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index