Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10 5/26/83; site ihuxq.UUCP Path: utzoo!linus!decvax!tektronix!uw-beaver!cornell!vax135!ariel!houti!hogpc!houxm!hocda!spanky!burl!we13!ihnp4!ihuxq!partha From: partha@ihuxq.UUCP Newsgroups: net.math Subject: combinatorial problem Message-ID: <267@ihuxq.UUCP> Date: Fri, 15-Jul-83 15:07:46 EDT Article-I.D.: ihuxq.267 Posted: Fri Jul 15 15:07:46 1983 Date-Received: Tue, 19-Jul-83 21:04:44 EDT Organization: BTL Naperville, Il. Lines: 62 I came across a problem while trying to prove fault coverage of algorithms for semiconductor RAM testing. The following is the problem: Given a nonempty set T where |T|=N, form any q (q less than N) nonempty subsets t1,t2,.....,tq of T where none of the subsets is disjoint with any other (i.e., ti intersection tj is not null for any i,j in 1 to q) and t1 U t2 U .... U tq = T. Does there exist a nonempty set R less than or equal to T such that |R intersection ti| = 0 or an even number for every ti in 1 to q? Two examples to illustrate the problem: a) Consider T={a,b,c}. Here |T|=3. For this case q could be 1 or 2. case q=1: Obviously t1={a,b,c}. There are three choices for R: R={a,b} or R={b,c} or R={c,a}. case q=2: There are several choices for t1,t2 and R here. Choice1: t1={a,b}, t2={b,c}, R={a,b,c}. Choice2: t1={a,c}, t2={b,c}, R={a,b,c}. Choice3: t1={b,c}, t2={a,b}, R={a,b,c}. Choice4: t1={a}, t2={a,b,c}, R={b,c}. ... etc ... b) Consider T={a,b,c,d}. Here |T|=4. q could be 1,2 or 3. case q=1: Again t1={a,b,c,d}. R={a,b,c,d} or R={a,b} or R={b,c} or R={c,d} or R={d,a} or R={a,c} or R={b,d}. case q=2: Choice1: t1={a,b}, t2={b,c,d}, R={c,d}. Choice2: t1={a,b,c}, t2={b,c,d}, R={b,c}. Choice3: t1={a}, t2={a,b,c,d}, R={b,c} or R={c,d} or R={b,d}. ... etc ... case q=3: Choice1: t1={a,b}, t2={b,c}, t3={a,b,d}, R={a,b,c}. Choice2: t1={a,b,c}, t2={c,d}, t3={a,b,c,d}, R={a,b}. Choice3: t1={a,b,c}, t2={b,c,d}, t3={c,d,a}, R={a,b,d}. ... etc ... My conjecture is that for any N, one can always find R satisfying the above conditions. Obviously enumeration is not the answer to prove this. I need an existence proof for this problem. Any help in this direction will be greatly appreciated. Thanks! P.S.: I was able to think of this problem in set theoretic notation. It may be easier to attack this problem if formulated otherwise (for instance, as a graph theory problem). Partha Raghavachari Bell Telephone Labs. Naperville, Ill. ihuxq!partha