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For positive integers d and n let fd(n) denote the maximum cardinality of a subset ofthe nd-grid f1; 2; : : : ; ngdwith distinct mutual euclidean distances. Improving earlierresults of Erdos and Guy, it will be shown that f2(n)  c \Delta n2=3and, for d  3, thatfd(n)  cd \Delta n2=3\Delta (ln n)1=3, where c; cd ? 0 are constants. Also improvements of lowerbounds of Erdos and Alon on the size of Sidon-sets in f12; 22; : : : ; n2g are given.Furthermore, it...

Point sets with distinct distances