Let α be the supremum of all δ such thatthere is a sequence Ann=1∞ of measurablesubsets of (0,1) with the property that each An has measureat least δ and for all n,m∈N, An∩ Am∩An+m=∅. For k∈N, let αk be thecorresponding supremum for finite sequences Ann=1k.We show that α=limk→∞αk andfind the exact value of αk for k≦41. In the process offinding these exact values, we also determine exactly the numberof maximal sum free subsets of {1,2,...,k} for k≦41. We alsoinvestigate the size of sets Axx∈S withAx∩ Ay∩ Ax+y=∅ where S is a subsemigroup of ((0,∞),+).
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机译：令α为所有δ的最大值，使得（0,1）的可测量子集有一个序列 n =1∞，其性质为每个An具有至少δ的值，并且对于所有n，m∈N，An∩Am ∩An+ m =∅。对于k∈N，令αk为有限序列 n = 1k的对应极值。我们证明α= limk→∞αk并找出k≤41的αk的精确值。在寻找这些精确值的过程中，我们还精确确定了{1,2，...，k}的最大无和子集的数量，其中k≤41。我们还用Ax∩Ay∩Ax + y =∅研究集合x∈S的大小，其中S是（（0，∞），+）的一个亚半群。
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