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Abstract |
Let n, s, and k be positive integers such that k ≥ 3, s ≥ 3 and n ≥ ks. An s-matching Ms in a k-uniform hypergraph is a set of s pairwise disjoint edges. The anti-Ramsey number ar(n, k, Ms) of an s-matching is the smallest integer c such that each edge-coloring of the n-vertex k-uniform complete hypergraph with exactly c colors contains an s-matching with distinct colors. The value of ar(n, k, Ms) was conjectured by Ozkahya and Young (Anti- Ramsey number of matchings in hypergraphs, Discrete Math., 313 (2013), 2359–2364) for all n ≥ sk and k ≥ 3. Frankl and Kupavskii (Two problems on matchings in set families - in the footsteps of Erdos and Kleitman, J. Combin. Theory ser. B, 138 (2019), 286–313) verified this conjecture for all n ≥ sk + (s − 1)(k − 1) and k ≥ 3. We aim to determine the value of ar(n, 3, Ms) for 3s ≤ n < 5s − 2 in this paper. Namely, we prove that if 3s < n < 5s − 2 and n is large enough, then ar(n, 3, Ms) = ex(n, 3, Ms−1) + 2. Here ex(n, 3, Ms−1) is the Turan number of an (s − 1)-matching. Thus this result confirms the conjecture of Ozkahya and Young for k = 3, 3s < n < 5s − 2 and sufficiently large n. For n = ks and k ≥ 3, we present a new construction for the lower bound of ar(n, k, Ms) which shows the conjecture by Ozkahya and Young is not true. In particular, for ¨ n = 3s, we prove that ar(n, 3, Ms) = ex(n, 3, Ms−1) + 5 for sufficiently large n.
This is a joint work with Mingyang Guo and Xing Peng. |
