If k is a positive real number, we say that a set S of real numbers is k-sum-free if there do not exist x, y, z in S such that x+y = kz. For k greater than or equal to 4 we find the essentially unique measurable k-sum-free subset of (0, 1] of maximum size.

If k is a positive integer, we say that a set A of positive integers is k-sum-free if there do not exist a, b, c in A such that a + b = kc. In particular we give a precise characterization of the structure of maximum sized k-sum-free sets in {1, . . . , n} for k ≥ 4 and n large.

In this paper we study recurrences concerning the combinatorial sum [n r ] m = ∑ k≡r (mod m) (n k ) and the alternate sum ∑ k≡r (mod m)(−1) (n k ) , where m > 0, n > 0 and r are integers. For example, we show that if n > m−1 then b(m−1)/2c ∑ i=0 (−1) (m− 1− i i )[n− 2i r − i ]

Let G be a graph. A zero-sum flow of G is an assignment of non-zero real numbers to the edges such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A zero-sum k-flow is a flow with values from the set {±1, . . . ,±(k − 1)}. It has been conjectured that every r-regular graph, r ≥ 3, admits a zero-sum 5-flow. In this paper we give an affirmativ...

For a sum of the form ∑ k F (n, k)G(n, k), we set up two systems of equations involving shifts of F (n, k) and G(n, k). Then we solve the systems by utilizing the recursion of F (n, k) and the method of undetermined coefficients. From the solutions, we derive linear recurrence relations for the sum. With this method, we prove many identities involving Bernoulli numbers and Stirling numbers.

Let K be a knot in the 3–sphere S3 , t(K) the tunnel number of K and K1#K2 the connected sum of two knots K1 and K2 , where t(K) is the minimal genus −1 among all Heegaard splittings which contain K as a core of a handle. Concerning the relationship between t(K1)+ t(K2) and t(K1#K2), we showed in Morimoto [2] that there are infinitely many tunnel number two knots K such that t(K#K′) is two agai...