It is shown that with probability 1-o(1)all eigenvalues belong to the above intervalI if μ=0, while in case μ>0 only the largest eigenvalueλ1 is outsideI, and λ1 asymptotically has a normal distribution with expectation (n−1)μ+v+(σ2/μ) and variance 2σ2 (bounded variance!).Expand

It is shown that the only optimal families, i.e., ∥F∥ = f(k, n, Σ) arise from the unique (11, 5, 4) or (12, 6, 5) Steiner-systems by a simple operation, called blowup.Expand

The validity of this conjecture is established for n ⩾ n 0 ( k), in a more general framework, when the excluded configuration is a fixed sunflower.Expand

There exists a pointset of cardinality at least 1.15 4 in E d such that all angles determined by the triples of are less than π/2. This disproves the old conjecture that | ≤ 2 d - 1.

It is shown that if F is a family of k-subsets of and n-set no two of which intersect in exactly l elements then for k ⩾ 2l + 2 and n sufficiently large F with equality holding if and only if F consists of all the k-sets containing a fixed (l + 1)-set.Expand

AbstractLetfr(n, k) denote the maximum number ofk-subsets of ann-set satisfying the condition in the title. It is proved that
$$f_1 (n,r(t - 1) + 1 +… Expand