On the number of groups of a given order

Letting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ?(n) and G(n) ≤ (log n)^c on a set of positive density. Letting F_k(x) denote the number of n ≤ x for which G(n) = k, it is shown that $\text{F}{}_2\text{(x) =}\text{O(x(}\text{log}{}_4\text{x)}\text{(}\text{log}{}_3\text{x)}{}^2\text{)}$, where log_rx denotes the r-fold iterated logarithm.     

On the number of homogeneous subgraphs of a graph

Let us defineG(n) to be the maximum numberm such that every graph onn vertices contains at leastm homogeneous (i.e. complete or independent) subgraphs. Our main result is exp (0.7214 log² n) ≧G(n) ≧ exp (0.2275 log² n), the main tool is a Ramsey—Turán type theorem. We formulate a conjecture what supports Thomason’s conjecture R(k, k)^1/k = 2.     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .     

On the average of the number of imaginary quadratic fields with a given class number

Let \(\mathcal {F}(h)\) be the number of imaginary quadratic fields with class number h. In this note, we improve the error term in Soundararajan’s asymptotic formula for the average of \(\mathcal {F}(h)\). Our argument leads to a similar refinement of the asymptotic for the average of \(\mathcal {F}(h)\) over odd h, which was recently obtained by Holmin, Jones, Kurlberg, McLeman and Petersen.     

On a loss system with a given number of customers

A one-server loss system with Poisson arrival stream and deterministic service times is considered conditional on the number of customers who appeared up to a givenT. This condition implies that the arrival times form a sample of the uniform distribution on (0,T]. We derive several characteristics of interest, such as the blocking probability at any given timet (0,T], the probability that exactlyi of the customers in (0,T] are served and, as a generalization, the distribution of the number of served customers arriving in any subinterval of (0,T].     

On the number of subgroups of given structure in a finitep-group

Supported in part by the Ministry of Science and Technology of Israel and the Rashi Foundation.     

On the number of elements of given order in a finitep-group

A. Kulakoff [9] proved that forp>2 the numberN _k =N _k (G) of solutions of the equationx ^{p k} =e in a non-cyclicp-groupG is divisible byp ^k+1. This result is a generalization of the well-known theorem of G. A. Miller asserting that the numberC _k =C _k (G) of cyclic subgroups of orderp ^k >p>2 is divisible byp. In this note we show that, as a rule: (1) ifk>1, thenN _k ≡0(modp ^k+p ); (2) ifk>2, thenC _k ≡0(modp ^p ). These facts are generalizations of many results from [1–5,8,9].     

On the number of certain subgraphs contained in graphs with a given number of edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphsG, H, letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also, forl≧0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We determineN(l, H) precisely for alll≧0 whenH is a disjoint union of two stars, and also whenH is a disjoint union ofr≧3 stars, each of sizes ors+1, wheres≧r. We also determineN(l, H) for sufficiently largel whenH is a disjoint union ofr stars, of sizess ₁≧s ₂≧…≧s _r>r, provided (s ₁−s _r)²<s ₁+s _r−2r. We further show that ifH is a graph withk edges, then the ratioN(l, H)/l ^k tends to a finite limit asl→∞. This limit is non-zero iffH is a disjoint union of stars.     

On the cardinality of power homogeneous compacta

We prove that if X is a power homogeneous compact space then |X|⩽2^{c(X)·π_χ(X)}. This generalizes similar results of Arhangel'skiı̆, van Douwen and Ismail. We apply this result to get new estimates for the cardinality of (power) homogeneous compacta satisfying some special conditions.     

On convex bodies containing a given number of lattice points

Given a convex body K in Euclidean space, a necessary and sufficient condition is established in order that for each n there exists a homothetic copy of K containing exactly n lattice points. Similar theorems are proved for congruent copies of K and for discrete sets other than lattices.     

On the number of vertices of given degree in a random graph

This note can be treated as a supplement to a paper written by Bollobas which was devoted to the vertices of a given degree in a random graph. We determine some values of the edge probability p for which the number of vertices of a given degree of a random graph G ∈ ??(n, p) asymptotically has a normal distribution.     

On the number of subgraphs of prescribed type of graphs with a given number of edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For a graphH=〈V(H),E(H)〉 and forS ⊂V(H) defineN(S)={x ∈V(H):xy ∈E(H) for somey ∈S}. Define alsoδ(H)= max {|S| − |N(S)|:S ⊂V(H)},γ(H)=1/2(|V(H)|+δ(H)). For two graphsG, H letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also forl>0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We investigate the asymptotic behaviour ofN(l, H) for fixedH asl tends to infinity. The main results are:Theorem A. For every graph H there are positive constants c ₁, c₂ such that {fx116-1}. Theorem B. If δ(H)=0then {fx116-2},where |AutH|is the number of automorphisms of H. (It turns out thatδ(H)=0 iffH has a spanning subgraph which is a disjoint union of cycles and isolated edges.) This paper forms part of an M.Sc. Thesis written by the author under the supervision of Prof. M. A. Perles from the Hebrew University of Jerusalem.     

On the possible number of elements of given order in a finite group

The main motivation of this paper is to introduce a problem of some combinatorial flavor about finite groups which seems to be new in the literature. Letk>1 be a fixed positive integer and denote byf(k, G) the number of elements of orderk in the groupG. We examine the setF(k)={f(k, G)| G a finite group}/{0}. We give a complete characterization ofF(k) if 4|k ork=6 and show some modest partial results for certain other values ofk. It seems to us that the question is surprisingly difficult even in such simple cases ask=3, which we investigate in detail. Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1901. Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1903.     

On the number of monic integer polynomials with given signature

In this paper, we show that the number of monic integer polynomials of degree \(d \ge 1\) and height at most H which have no real roots is between \(c_1H^{d-1/2}\) and \(c_2 H^{d-1/2}\), where the constants \(c_2>c_1>0\) depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying \(0 \le s < d/2\) we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to \(\lambda (d,s)H^{d}\) as \(H \rightarrow \infty \). The constants \(\lambda (d,s)\) are all positive and come from a recent paper of Bertók, Hajdu, and Peth?. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.