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@fedja You are correct that a product of the sort that you describe would suffice to answer my question. I searched for such products. If I remember correctly there were none with coefficients of the required sort that went beyond x^19 or thereabouts. This was a relatively easy computation. But the search space for what I am looking for is much larger.
@ Gerry Myerson. I sent Graham a set of 23 intervals which satisfy the conditions of the problem. This may explain his statement that s(1)>=23. With further calculation I found a set with 25 intervals. So s(1)>=25.
I added the quotation of Steinhaus as suggested in one of the comments. I removed a comment about being unable to resolve a question of formatting since that prolem had been taken care of in a previous edit.
