Zero-Sum Km Over Z and the Story of K4.

We prove the following results solving a problem raised by Caro and Yuster (Graphs Comb 32:49–63, 2016). For a positive integer m ≥ 2 , m ≠ 4 , there are infinitely many values of n such that the following holds: There is a weighting function f : E (K n) → { - 1 , 1 } (and hence a weighting function f : E (K n) → { - 1 , 0 , 1 } ), such that ∑ e ∈ E (K n) f (e) = 0 but, for every copy H of K m in K n , ∑ e ∈ E (H) f (e) ≠ 0 . On the other hand, for every integer n ≥ 5 and every weighting function f : E (K n) → { - 1 , 1 } such that | ∑ e ∈ E (K n) f (e) | ≤ n 2 - 2 h (n) , where h (n) = (n + 1) if n ≡ 0 (mod 4) and h (n) = n if n ≢ 0 (mod 4), there is always a copy H of K 4 in K n for which ∑ e ∈ E (H) f (e) = 0 , and the value of h(n) is sharp. [ABSTRACT FROM AUTHOR]