Rainbow matchings in properly-colored hypergraphs

A hypergraph $H$ is properly colored if for every vertex $v\in V(H)$, all the edges incident to $v$ have distinct colors. In this paper, we show that if $H_{1}$, \cdots, $H_{s}$ are properly-colored $k$-uniform hypergraphs on $n$ vertices, where $n\geq3k^{2}s$, and $e(H_{i})>{{n}\choose {k}}-{{n-s+1}\choose {k}}$, then there exists a rainbow matching of size $s$, containing one edge from each $H_i$. This generalizes some previous results on the Erd\H{o}s Matching Conjecture.

proved it for sufficiently large s. The k = 3 case was finally settled by Frankl [8]. For general k, a short calculation shows that when s ≤ n k+1 , we always have n k − n−s+1 k > ks−1 k . For this range, the second construction is believed to be optimal. Erdős [4] proved the conjecture for n ≥ n 0 (k, s).
Bollobás, Daykin and Erdős [2] proved the conjecture for n > 2k 3 (s − 1). Huang, Loh and Sudakov [12] improved it to n ≥ 3k 2 s, which was further improved to n ≥ 3k 2 s/log k by Frankl, Luczak and Mieczkowska [10]. On the other hand, in an unpublished note, Füredi and Frankl proved the conjecture for n ≥ cks 2 , Frankl [7] improved all the range above to n ≥ (2s − 1)k − s + 1. Currently the best range is n ≥ 5 3 sk − 2 3 s by Frankl and Kupavskii [9]. In this paper, we consider a generalization of Erdős Matching Conjecture to properly-colored hypergraphs. A hypergraph H is properly colored if for every vertex v ∈ V (H), all edges incident to v are colored differently. A rainbow matching in a properly-colored hypergraph H is a collection of vertex disjoint edges with pairwise different colors. The size of a rainbow matching is the number of edges in the matching. The rainbow matching number, denoted by ν r (H), is the maximum size of a rainbow matching in H. Motivated by the Erdős Matching Conjecture, we consider the following problem: how many edges can appear in a properly-colored k-uniform hypergraph H such that its rainbow matching number satisfies ν r (H) < s ≤ n k ? In fact, it is called Rainbow Turán problem and is well studied in [13]. Note that here if we let H be rainbow, that is, every edge of H receives distinct colors, then we obtain the original Erdős Matching Conjecture.
More generally, let H 1 , . . . , H s be properly-colored k-uniform hypergraphs on n vertices, a rainbow matching of size s in H 1 , . . . , H s is a collection of vertex disjoint edges e 1 , . . . , e s with pairwise different colors, where e 1 ∈ E(H 1 ), . . . , e s ∈ E(H s ). For simplicity, we call it an s-rainbow matching. Then what is the minimum M , such that by assuming e(H i ) > M for every i, it guarantees the existance of an s-rainbow matching?
In this paper, we prove the following result, which generalizes Theorem 1.2 and Theorem 3.3 of [12].

Preliminary results
In this section, we list some preliminary results about "rainbow" hypergraphs, which is a special case of properly-colored hypergraphs. In the next section, we will prove our main theorem with the help of these results. A hypergraph H is rainbow if the colors of any two edges in E(H) are different. From now on, when we say an edge e is disjoint from a collection of edges, it means that not only e is vertex-disjoint from those edges, but it also has a color different from the colors of all these edges. We start by the following lemma for graphs. Note that here although each G i is rainbow, a color may appear in more than one G i 's.
. . , G s be rainbow graphs on n vertices. If n ≥ 5s and e(G i ) > n 2 − n−s+1 2 , then there exists an s-rainbow matching in G 1 , . . . , G s .
Proof. We do induction on s. The base case s = 1 is trivial. For every vertex v ∈ V (G i ) and there exists an edge e in G i which contains v and disjoint from the edges of the (s − 1)-rainbow matching, which produces an s-rainbow matching. Hence we may assume that the maximum degree of each G i is at most 3(s − 1). Now pick an arbitrary edge uv in G 1 . Assume the color of uv is c(uv). Then we delete the vertices u, v and the edge colored by c(uv) in G 2 , . . . , G s . Denote the resulting graphs by G ′ 2 , . . . , G ′ s . We can see that when n ≥ 5s, . By induction on s, there exists an (s − 1)-rainbow matching in the graphs G ′ 2 , . . . , G ′ s . Taking these s − 1 edges with the edge uv, we obtain an s-rainbow matching in G 1 , . . . , G s . Proof. We do induction on both k and s. According to Lemma 2.1, the case k = 2 holds for every s and n ≥ 5s. And for every k, the case s = 1 is trivial. We first consider the situation when some By inductive hypothesis for the case (n − 1, k, s − 1), there exists an then there exists an edge e in E(H i ) which contains v and disjoint from the edges of the (s − 1)-rainbow matching, which produces an s-rainbow matching. Hence we may assume that the maximum degree in each hypergraph H i is at most k(s − 1) n−2 k−2 + s − 1.
By induction on s, we know that for every i there exists an (s − 1)-rainbow matching in the hypergraphs {H j } j =i , spanning k(s − 1) vertices. If for some i, the s-th largest degree of H i is at most 2(s − 1) n−2 k−2 + s − 1, then the sum of degrees of these k(s − 1) vertices in H i is at most

Main Theorem
In this section we prove our main result, Theorem 1.2, using induction and Lemma 2.2.
Proof. We split our proof into two cases. On the other hand, the maximum degree of the subgraph of H i by deleting these k(s − 1) vertices is at most s − 1, otherwise, we can find an s-rainbow matching. Since n ≥ 3k 2 s, we have which guarantees the existence of an edge in H i disjoint from the previous (s − 1)-rainbow matching in {H j } j =i , which produces an s-rainbow matching. So we may assume that each H i contains at least s vertices with degree above 2(s − 1) n−2 k−2 + s − 1. Now we may greedily select distinct vertices v i ∈ V (H i ), such that for each 1 ≤ i ≤ s, the degree In this short note, we propose a generalization of the Erdős hypergraph matching conjecture to finding rainbow matchings in properly-colored hypergraphs, and prove Theorem 1.2 for s < n/(3k 2 ).
The following conjecture seems plausible. Recall that for the special case when each H i is identical and rainbow, Frankl and Kupavskii [9] were able to verify it for C = 5/3. However the proof relies on the technique of shifting, while the property of a hypergraph being properly colored may not be preserved under shifting.
It is tempting to believe that Erdős Matching Conjecture can be extended to properly-colored hypergraphs for the entire range of s, that is, once the number of edges in each hypergraph exceeds the maximum of n k − n−s+1 k and ks−1 k , then one can find an s-rainbow matching. However this is false in general, a simple construction is by taking s = 2 and n = 2k. The maximum of these two expressions is 2k−1 k , while one can let H 1 be a rainbow K k 2k with an edge coloring c 1 , and H 2 be on the same vertex set with edge coloring c 2 , such that c 2 (e) = c 1 ([2k] \ e). Then clearly each H i contains 2k k > 2k−1 k edges and there is no 2-rainbow matching. It would be interesting to find constructions for s close to n/k, and formulate a complete conjecture for properly-colored hypergraphs.