In this paper, we consider a randomized greedy algorithm for independent sets in r-uniform d-regular hypergraphs G on n vertices with girth g. By analyzing the expected size of the independent sets generated by this algorithm, we show that (G) ≥ (f ( , r) − (g, , r))n, where (g, , r) converges to 0 as g → ∞ for fixed d and r, and f (d, r) is determined by a differential equation. This extends earlier results of Garmarnik and Goldberg for graphs [8]. We also prove that when applying this algorithm to uniform linear hypergraphs with bounded degree, the size of the independent sets generated by this algorithm concentrate around the mean asymptotically almost surely.

Given a finite simple graph G, an odd cover of G is a collection of complete bipartite graphs, or bicliques, in which each edge of G appears in an odd number of bicliques and each non-edge of G appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of G by b2(G) and prove that b2(G) is bounded below by half of the rank over F2 of the adjacency matrix of G. We show that this lower bound is tight in the case when G is a bipartite graph and almost tight when G is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from b2(G).Babai and Frankl (1992) proposed the "odd cover problem," which in our language is equivalent to determining b2(Kn). Radhakrishnan, Sen, and Vishwanathan (2000) determined b2(Kn) for an infinite but density zero subset of positive integers n. In this paper, we determine b2(Kn) for a density 3/8 subset of the positive integers.

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