Positive linear combination free families ∗

A family of subsets of [n] is positive linear combination free if the characteristic vector of neither edge is the positive linear combination of the characteristic vectors of some other ones. We construct a positive linear combination free family which contains (1−o(1))2 subsets of [n] and we give tight bounds on the o(1)2 term. The problem was posed by the late Levon Khachatrian and the result has geometric consequnces. 1 Positive linear combination free families The following question was posed by Levon Khachatrian on EuroComb01 in Barcelona. How many edges may a hypergaph on n vertices contain such that the characteristic vector of neither edge is the positive linear combination of the characteristic vectors of some other ones? He added that they worked on this problem with Rudolf Ahlswede, and got a construction with (1/2 + c)2n sets. They wanted to know if such a family may contain almost all edges or significantly less. Here we give an explicit cosntruction for such a family which contains (1 − o(1))2n edges and tight bounds for the o(1)2n term. Let [n] = {1, . . . , n}. The characteristic vector of A ⊆ [n] is the vector A in {0, 1}n which has 1 in the ith coordinate iff i ∈ A. (We use the same notation for sets and characteristic vectors.) A is the ∗2000 Mathematics Subject Classification: 05D05, 52B05 †Department of Mathematics, University of Illinois, Urbana, IL61801, USA, and Rényi Institute of Mathematics of the Hungarian Academy of Sciences, Budapest, P. O. Box 127, Hungary-1364. Email: [email protected]. Research supported in part by the Hungarian National Science Foundation under grant OTKA T 032452, and by the National Science Foundation under grant 1-5-29066 NSF DMS 00-70312. ‡Computer and Automation Research Institute of the Hungarian Academy of Sciences, Budapest, P. O. Box 63, Hungary-1518. Email: [email protected]. Research supported in part by OTKA Grants T029074, T030059, T038198, Bolyai Fellowship, and KFKI-ISYS Datamining Lab.