ON PA RWISE BALANCED BLOCK DESIGNS WITH THE SIZES OF BLOCKS AS UNIFORM AS POSSIBLE P. ERDŐS and J. LARSON
نویسندگان
چکیده
Let 1(~j = n, Ai C r5, 1-i -T„ is a partially balanced block design, JAIJ -• • s ~A j . The authors prove that there is such a design for which lA i l = n li2 + O(n lrz`) for some c > 0 . If certain plausible assumptions on the difference of consecutive primes are made, then the above inequality can be improved to lA i l = nV 2 +O((log n)Z ) . It is true that there is a design with HAll > n"'c ? This challenging problem is left open .
منابع مشابه
Extremal Problems 2or Pairwise Balanced Designs
1 . Introduction In a previous paper [2] the senior author discussed many problems concerning block designs . Some of these have since been settled . Szemerédi and Trotter [5] showed, for example, that the number of pairwise balanced designs which can be obtained geometrically on n points in the plane with the blocks being the lines joining these points is at most exp(c/) . In [1] Colbourn, Phe...
متن کاملIncomplete Block Designs
1. Introduction These designs were introduced by Yates in order to eliminate heterogeneity to a greater extent than is possible with randomized blocks and Latin squares when the number of treatments is large. The precision of the estimate of a treatment effect depends on the number of replications of the treatment-the larger the number of replications, the more is the precision. Similar is the ...
متن کاملPairwise balanced designs with block sizes 4 and 8
Let K be a set of positive integers. A pairwise balanced design (PBD) of index unity B(K,l;v) is a pair (X/~) where X is a v-set (of points) and B is a collection of subsets of X (called blocks) with sizes from K such that every pair of distinct points of X is contained in exactly one block of~. A necessary condition for the existence of a PBD B({4,8},I;v) is v-0 or l(mod 4). It is shown that t...
متن کاملOn construction of partially balanced n-ary designs
A new method of constructing a series of partially balanced ternary (PBT) designs is presented. In the method, we have added the corresponding rows of incidence matrices of a BIB design and a PBIB design, both obtained from single initial block with at least one element in common between them. The BIB and PBIB designs above were obtained by method of differences. We have also constructed PBT de...
متن کاملPairwise balanced designs from nite elds
Developing previous work by R.C. Bose, R.M. Wilson, M. Greig and the author, we give a general construction for di erence families over nite elds. All the DFs constructed before were generated by a single initial base block so that they were uniform. Here we also consider DFs generated by two or more initial base blocks. These DFs give rise to pairwise balanced designs which of course are balan...
متن کامل