SECTIONAL TRANSACTIONS.— A*. 333 



where 



M m [(k - 2) log 2 + log (k-2)~ log k~[ 



(2) m -l log*-log(*-i) J' 



and [#] denotes the integral part of x. In particular G(4) <C 17. This was 

 recently proved by Davenport and Heilbronn and simultaneously by me, 

 the method being essentially one of Winogradoff's refinements of the classical 

 Hardy-Littlewood method. In this paper the same method is applied to 

 the general case. , 



It follows from (1) and (2) that G(s) ^ 29 and G(6) ^ 42. 



Dr. P. Erdos. — Note on some properties of sequences of integers (12.30). 



Let a x < a 2 < . . ■ < a x < n be a sequence of positive integers such 

 that no ai is contained in the product of two other a's of the sequence. 

 Then 



"<"« + (jR^)> 



this error term is the best possible. 



The proof is more intellegible if I first prove only that 



x < iz(n) + 2 ni 



In this case the proof is based on the lemma : 



Any integer m ^ n may be written in the form b% cj where b% denotes any 

 integer not exceeding n$ or any prime of the interval (n$, n), and cj any 

 integer not exceeding n«, 



To prove for * the more precise inequality we need a refined and rather 

 complicated form of the lemma : 



Now let a 1 < <x 2 < . . . a y ^ n be another sequence of positive integers 

 such that the products a, ay are all different. Then 



y < n(n) + O(m') 



The proof is based on our previous lemma. 



((log n)i. ) 



Here the error term cannot be better than O 



Friday, September 11. 



Dr. B. Kauffmann. — Some recent results in general topology (11. 10). 



Modern topology has developed from two originally independent subjects : 

 combinatorial topology or analysis situs (in the sense of H. Poincare) and 

 the general theory of abstract spaces. The following stages of this develop- 

 ment are considered in the first part of the lecture : 



(1) Discovery of the combinatorial nature of the problems of general 

 topology. J. W. Alexander's duality theorem and its generalisation by 

 Lefschetz, Pontrjagin and others. 



(2) Theory of simplicial approximations and transformations by Brouwer, 

 Alexandroff and others. H. Hopf's fundamental result on essential trans- 

 formations of complexes. 



(3) Theory of dimension. 



(4) P. Alexandroff's work on the internal structure of general spaces. 

 This last mentioned work of Alexandroff is the starting point of a new 



theory, which is outlined in the second part of the paper : 



