László Pyber

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László Pyber (born 8 May 1960 in Budapest) is a Hungarian mathematician. He works in combinatorics and group theory. He is a researcher at the Alfréd Rényi Institute of Mathematics, Budapest.He received the title the Doctor of Science from the Hungarian Academy of Sciences (1998). He won the Academics Prize (2007).

Main results

He proved the conjecture of Paul Erdős and Tibor Gallai, that the edges of any simple graph with n vertices can be covered with at most n-1 circuits and edges.
He proved the following conjecture Paul Erdős. Any graph with n vertices and its complement can be covered with n²/4+2 cliques.
He proved a clog²n bound to the size of a minimal base of a primitive permutation group of degree n not containing A_n.
He gave the following estimate of the number of groups of order n. If the prime power decomposition of n is n=p₁^g₁ ⋯ p_k^g_k and μ=max(g₁,...,g_k), then the number of nonisomporphic n-element groups is at most

n^{(\frac{2}{27}+o(1))\mu^2}.

Łuczak and Pyber proved the following conjecture of McKay. For every, ε>0 there is a number c such that for all sufficiently large n, c randomly chosen elements generate the symmetric group S_n with probability greater than 1-ε.
A result also proved by Łuczak and Pyber states that almost every element of S_n does not belong to a transitive subgroup different from S_n and A_n (conjectured by Cameron).
Solving a problem of subgroup growth he proved that for every nondecreasing function g(n)≤log(n) there is a residually finite group generated by 4 element, whose growth type is $n^{g(n)}$ .

L. Pyber: An Erdős-Gallai conjecture, Combinatorica, 5(1985), 67–79.
L. Pyber: Clique covering of graphs, Combinatorica, 6(1986), 393–398.
L. Pyber: Enumerating finite groups of given order, Annals of Mathematics, (2), 137(1993), 203–220.
L. Pyber: On the orders of doubly transitive permutation groups, elementary estimates, J. Combin. Theory, (A), 62(1993), 361–366.
L. Pyber: Groups of intermediate subgroup growth and a problem of Grothendieck, Duke Math. J., 121(2004), 169–188.
A. Jaikin-Zapirain, L. Pyber: Random generation of finite and profinite groups and group enumeration, to appear in Annals of Mathematics.