• Almost Odd Random Sum-Free Sets 

      Calkin, Neil J.; Cameron, P. J. (Georgia Institute of Technology, 1995-07)
      We show that if S_1 is a strongly complete sum-free set of positive integers, and if S_0 is a finite sum-free set, then with positive probability a random sum-free set U contains S_0 and is contained in S_0\cup S_1. As ...
    • Counting Sets of Integers, No k of Which Sum to Another 

      Calkin, Neil J.; Taylor, Angela C. (Georgia Institute of Technology, 1995-07)
      We show that for every k greater or equal than 3 the number of subsets of {1,2,...,n} containing no solution to x_1 + x_2 + ... + x_k = y, where the x_i need not be distinct, is at most c2^{\alpha n}, where \alpha = (k-1)/k.
    • A Curious Binomial Identity 

      Calkin, Neil J. (Georgia Institute of Technology, 2009-12-07)
      In this note we shall prove the following curious identity of sums of powers of the partial sum of binomial coefficients.
    • Dependent Sets of Constant Weight Binary Vectors 

      Calkin, Neil J. (Georgia Institute of Technology, 1995-07)
      We determine lower bounds for the number of random binary vectors, chosen uniformly from vectors of weight k, needed to obtain a dependent set.
    • New Ramsey Bounds from Cyclic Graphs of Prime Order 

      Calkin, Neil J.; Erdös, Paul; Tovey, Craig A. (Georgia Institute of Technology, 1995-09)
      We present new explicit lower bounds for some Ramsey numbers. All the graphs are cyclic, and are on a prime number of vertices. We give a partial probabilistic analysis which suggests that the cyclic Ramsey numbers grow ...
    • The Number of Independent Sets in a Grid Graph 

      Calkin, Neil J.; Wilf, Herbert S. (Georgia Institute of Technology, 1995-07)
    • On a Class of Aperiodic Sum-Free Sets 

      Calkin, Neil J.; Erdös, Paul (Georgia Institute of Technology, 1994-10)
      We show that certain natural aperiodic sum-free sets are incomplete that is that there are infinitely many n not in S which are not a sum of two element of S.
    • Some Conditions on Periodicity for Sum-Free Sets 

      Calkin, Neil J.; Finch, Steven R. (Georgia Institute of Technology, 1995-07)
      Cameron has introduced a natural bijection between the set of one way in nite binary sequences and the set of sum-free sets (of positive integers), and observed that a sum-free set is ultimately periodic only if the ...