Péter Pál Pach

A NEW OPERATION ON PARTIALLY ORDERED SETS

Abstract Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and the second type are permutations induced by so-called rotations. In this paper we introduce rotations for finite posets, which can be seen as the poset counterpart of Seidel-switch for finite graphs. We analyze some of their combinatorial properties, and investigate in particular the question of when two finite posets are rotation-equivalent. We moreover give an explicit combinatorial construction of a rotation of the random poset whose image is again isomorphic to the random poset. As a corollary of our results on rotations of finite posets, we obtain that the group of rotating permutations of the random poset is the automorphism group of a homogeneous structure in a finite language.

ON FREE ALGEBRAS IN VARIETIES GENERATED BY ITERATED SEMIDIRECT PRODUCTS OF SEMILATTICES

We present a new solution of the word problem of free algebras in varieties generated by iterated semidirect products of semilattices. As a consequence, we provide asymptotical bounds for free spectra of these varieties. In particular, each finite -trivial (and, dually, each finite -trivial) semigroup has a free spectrum whose logarithm is bounded above by a polynomial function.

The step Sidorenko property and non-norming edge-transitive graphs

Sidorenkos Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a quasirandom multipartite graph minimizes the density of H among all graphs with the same edge densities between its parts; this property is called the step Sidorenko property. We show that many bipartite graphs fail to have the step Sidorenko property and use our results to show the existence of a bipartite edge-transitive graph that is not weakly norming; this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150].

On infinite multiplicative Sidon sets

We prove that if

Multiplicative bases and an Erdős problem

A

Remarks On Commuting Graph of a Finite Group

is an infinite multiplicative Sidon set, then

THE AUTOMORPHISM GROUP OF COMMUTING GRAPH OF A FINITE GROUP

\liminf\limits_{n\to \infty}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n)^3}} 0

Progression-free sets in Z_4^n are exponentially small

.

Reducts of the random partial order

In this paper we investigate how small the density of a multiplicative basis of order h can be in {1,2,...,n} and in ℤ+. Furthermore, a related problem of Erdős is also studied: How dense can a set of integers be, if none of them divides the product of h others?

Progression-free sets in \mathbb Z_4^n are exponentially small

Abstract The commuting graph Δ ( G ) of a group G is the graph whose vertex set is the group and two distinct elements x and y being adjacent if and only if x y = y x . In this paper the automorphism group of this graph is investigated. We observe that Aut( Δ ( G ) ) is a non-abelian group such that its order is not prime power and square-free.