2026/08/02 by Cheng Chi, Jialin He, Quan Sun
Mathematics · #math.CO
arxiv created 2026/08/02 · arxiv updated 2026/08/04
Let KN(r) denote the N-vertex complete r-uniform hypergraph. For an r-uniform hypergraph H and an integer k≥2, the k-color Ramsey number R(H,k) is the least integer N such that every k-edge-coloring of KN(r) contains a monochromatic copy of H. When k|\esize(H), the zero-sum Ramsey number R(H,\mathbb Zk) is the least integer N such that every edge-labeling of KN(r) by elements of \mathbb Zk contains a copy of H whose edge labels sum to 0 in \mathbb Zk. We settle two conjectures and a problem concerning these two Ramsey numbers. First, Caro and Provstgaard proposed exact values for the zero-sum Ramsey numbers over \mathbb Z2 of delta-systems with an even number of edges. We determine these numbers and thereby prove their conjecture. Second, for a forest F with m edges, let tF denote the disjoint union of t copies of F. Caro conjectured that R(tF,\mathbb Zmt)=R(tF,2) for all sufficiently large t. We show that this conjecture does not hold for double stars. Caro also asked whether there exists a tree T with m edges such that R(T,\mathbb Zm)>R(T,2). We answer this question affirmatively by constructing an infinite family of such trees.