Let k be a positive integer. A sequence s 1 , s 2 , ź , s m over an n -element A alphabet is a packing k -radius sequence, if for all pairs of indices ( i , j ) , such that 1 ź i < j ź m and j - i ź k , the sets { s i , s j } are pairwise different 2 -element subsets of A . Let g k ( n ) denote the length of a longest k -radius sequence over A . We give a construction demonstrating that for every k = ź c n α ź , where c and α are fixed reals such that c 0 and 0 ź α < 1 2 , g k ( n ) = n 2 2 k ( 1 - o ( 1 ) ) . For a constant k we show that g k ( n ) = n 2 2 k - O ( n 1.525 ) . Moreover, we prove an upper bound for g k ( n ) that allows us to show that g k ( n ) = n ( 1 + o ( 1 ) ) for every k = ź c n α ź , where c 0 and 1 2 < α < 1 .
['Zbigniew Lonc', 'Miroslaw Truszczynski']
Packing analogue of k -radius sequences