Zoltán Füredi

The eigenvalues of random symmetric matrices

AbstractLetA=(aij) be ann ×n matrix whose entries fori≧j are independent random variables andaji=aij. Suppose that everyaij is bounded and for everyi>j we haveEaij=μ,D2aij=σ2 andEaii=v.E. P. Wigner determined the asymptotic behavior of the eigenvalues ofA (semi-circle law). In particular, for anyc>2σ with probability 1-o(1) all eigenvalues except for at mosto(n) lie in the intervalI=(−c√n,c√n).We show that with probability 1-o(1)all eigenvalues belong to the above intervalI if μ=0, while in case μ>0 only the largest eigenvalue λ1 is outsideI, and

Onr-Cover-free Families

\lambda _1 = \frac{{\Sigma _{i,j} a_{ij} }}{n} + \frac{{\sigma ^2 }}{\mu } + O\left( {\frac{I}{{\sqrt n }}} \right)

Families of Finite Sets in Which No Set Is Covered by the Union of Two Others

i.e. λ1 asymptotically has a normal distribution with expectation (n−1)μ+v+(σ2/μ) and variance 2σ2 (bounded variance!).

Maximum degree and fractional matchings in uniform hypergraphs

Almost all combinatorial question can be reformulated as either a matching or a covering problem of a hypergraph. In this paper we survey some of the important results.

ON THE NUMBER OF HALVING PLANES

A very short proof is presented for the almost best upper bound for the size of anr-cover-free family overnelements.

Point Selections and Weak ε-Nets for Convex Hulls

Let <italic>S</italic> ⊂ R<supscrpt>3</supscrpt> be an <italic>n</italic>-set in general position. A plane containing three of the points is called a halving plane if it dissects <italic>S</italic> into two parts of equal cardinality. It is proved that the number of halving planes is at most <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2.998</supscrpt>). As a main tool, for every set <italic>Y</italic> of <italic>n</italic> points in the plane a set <italic>N</italic> of size <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>4</supscrpt>) is constructed such that the points of <italic>N</italic> are distributed almost evenly in the triangles determined by <italic>Y</italic>.

A new generalization of the Erdős-Ko-Rado theorem

Abstract Let f k ( n ) denote the maximum of k -subsets of an n -set satisfying the condition in the title. It is proven that f 2 t − 1 (n) ⩽ f 2t (n + 1) ⩽ ( t n ) ( t 2t−1 ) with equalities holding iff there exists a Steiner-system S ( t , 2 t − 1, n ). The bounds are approximately best possile for k ⩽ 6 and of correct order of magnitude for k >/ 7, as well, even if the corresponding Steiner-systems do not exist. Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family (i.e., the nonuniform case).

Graphs without quadrilaterals

AbstractLet ℋ be a family ofr-subsets of a finite setX. SetD(ℋ)=