Combinatorial analysis (nonnegative matrices, algorithmic problems)

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“…In other words, the Erdös-Hanani hypothesis says that asymptotically exact coverings and packings exist for all fixed l < k. At the same time, they proved ( 23) and ( 24) for l = 2 and all fixed k and for l = 3 in the case k = p or k = p + 1, where p is a power of a prime. It is well known that for fixed l and k, (23) is equivalent to (24). For the first time, (23) was proved for l = k − 1 and k = o( √ n), and ( 24) was proved for l = k − 1 and k = o(n) ( [9]).…”

“…The question we consider in this section is to find a "threshold function" k(n) for the existence of asymptotically exact packings and coverings for a slow growth of l (with increase in n). This means that (23) or (24) is true for all k < c 0 k(n) and is not true for k > c 1 k(n) for any constraints c 0 < 1, with c 1 > 1 and a sufficiently large n. It turns out that asymptotically exact coverings exist for all k = o(n) for a slow growth of l as n → ∞. On the other hand, we demonstrate that k(n) = √ n is a threshold function for the existence of asymptotically exact packings.…”