Erdős posed the problem of finding conditions on a graph G that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general cases. Let δ r be the least number so that any graph G on n vertices with minimum degree δ r n has the property P r−1 (G) = K r f (G), where P r−1 (G) is the largest number of edges in an (r − 1)-partite subgraph and K r f (G) is the largest number of edges in a K r -free subgraph. We show that 3r−4 3r−1 < δ r ≤ 4(3r−7)(r−1)+1 4(r−2)(3r−4) when r ≥ 4. In particular, δ 4 ≤ 0.9415.
Define the superball with radius r and center 0 in R n to be the setwhich is a generalization of ℓ p -balls. We give two new proofs for the celebrated result that for 1 < p ≤ 2, the translative packing density of superballs in R n is Ω(n/2 n ). This bound was first obtained by Schmidt, with subsequent constant factor improvement by Rogers and Schmidt, respectively. Our first proof is based on the hard superball model, and the second proof is based on the independence number of a graph. We also investigate the entropy of packings, which measures how plentiful such packings are.
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