Turán problems for vertex-disjoint cliques in multi-partite hypergraphs
Introduction
An -uniform hypergraph, or simply an -graph, is a hypergraph whose edges have exactly vertices. For an -graph , let be the vertex set of and the edge set of . An -graph is called -free if does not contain any copy of as a subgraph. For two -graphs and , the Turán number is the maximum number of edges of an -free subgraph of . Denote by the complete -graph on vertices. A copy of in an -graph is also called a -clique of . Let denote the -graph consisting of vertex-disjoint copies of . If , then represents a matching of size . Let be integers and be disjoint vertex sets with for each . A complete -partite -graph on vertex classes , denoted by or , is defined to be the -graph whose edge set consists of all the -element subsets of such that for all . An -graph is called an -partite -graph on vertex classes if is a subgraph of . For , we often write and instead of and . Let denote the set and denote the set for .
Turán-type problems were first considered by Mantel [15] in 1907, who determined . In 1941, Turán [17] showed that the balanced complete -partite graph on vertices, called the Turán graph and denoted by , is the unique graph that maximizes the number of edges among all -free graphs on vertices. Since then, Turán numbers of graphs and hypergraphs have been extensively studied. However, even though lots of progress have been made, most of the Turán problems for bipartite graphs and for hypergraphs are still open. Specifically, none of the Turán numbers with has yet been determined, even asymptotically. We recommend the reader to consult [13], [16] for surveys on Turán numbers of graphs and hypergraphs.
Many problems in additive combinatorics are closely related to Turán-type problems in multi-partite graphs and hypergraphs. Recently, Turán problems in multi-partite graphs have received a lot of attention, see [3], [5], [12]. The following result, which is attributed to De Silva, Heysse and Young, determines .
Theorem 1.1 For and ,
Since it seems that their preprint has not been published online, we present a proof of Theorem 1.1 in the Appendix for the completeness of the paper. In [5], De Silva, Heysse, Kapilow, Schenfisch and Young determined .
Theorem 1.2 [5] For and ,
In this paper, we consider three Turán-type problems for disjoint cliques in -partite -graphs. Let be integers. For any , denote by . Define and
Theorem 1.3 For , and , if for ; for and for , then
It should be mentioned that the problem in Theorem 1.3 can be viewed as a multi-partite version of the Erdős matching conjecture, which states that and is still open when is close to , see [4], [6], [8], [9], [10] for recent progress. The lower bound in Theorem 1.3 follows from the following construction. Let be an -partite -graph on vertex classes with sizes , respectively. Let be a -element subset of . An edge of forms an edge of if and only if . It is easy to see that is -free. Otherwise, if has a matching of size , then we have since each edge of contains a vertex in .
As our second main result, we use a probabilistic argument to determine .
Theorem 1.4 For , and ,
The lower bound in Theorem 1.4 follows from the following construction. Let be an -partite -graph on vertex classes with sizes , respectively. Let be an -element subset of and let be obtained by deleting all the edges of from . It is easy to see that is -free. Otherwise, if there are vertex-disjoint copies of in , then we have since each copy of in contains a vertex in .
We also consider the generalized Turán problem in multi-partite graphs. Let denote the maximum number of copies of in an -free subgraph of . The first result of this type is due to Zykov [18], who showed that the Turán graph also maximizes the number of -cliques in an -vertex -free graph for . Recently, Alon and Shikhelman [2] determined asymptotically for any with chromatic number . Precisely, they proved that where denotes the number of -cliques in the Turán graph . Later, the error term of this result was further improved by Ma and Qiu [14].
In this paper, we also study the maximum number of -cliques in a -free subgraph of . By the same probabilistic argument as in the proof of Theorem 1.4, we obtain the following result.
Theorem 1.5 For , and ,
Note that for , and arbitrary , the Turán number is determined by Theorem 1.5. Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szabó [11], we also determine for and sufficiently larger than .
Theorem 1.6 For , , and , if , then
The lower bounds in Theorem 1.5, Theorem 1.6 follow from the same construction as follows. Let be an -partite graph on , which are of sizes , respectively. Let be an -element subset of . Then is obtained by deleting all the edges of from . It is easy to see that is -free. Otherwise, if there are vertex-disjoint copies of in , then we have since each copy of in contains a vertex in .
The rest of the paper is organized as follows. We will prove Theorem 1.3 in Section 2. In Section 3, we prove Theorem 1.4. In Section 4, we prove Theorem 1.5, Theorem 1.6.
Section snippets
Turán number of in -partite -graphs
In this section, we prove Theorem 1.3. First, we consider the case , which is the base case for other results in this paper. Aharoni and Howard [1] determined the maximum number of edges in a balanced -partite -graph that is -free. By the same argument, we prove the following result:
Lemma 2.1 For any integers ,
Proof We shall partition the edge set of into matchings of size each. Let for and
Turán number of in -partite -graphs
In this section, we generalize the result of [5] to -graphs by using a probabilistic argument. The following lemma will be useful for us.
Lemma 3.1 Assume that , and let be a linear programming model as follows: Let be the integral part of and . Then is the optimal value of .
Proof Suppose to the contrary that there exists a feasible solution to such that Since is a feasible
The number of -cliques in -partite graphs
In this section, we first determine for the case . Then, by utilizing a result on rainbow matchings, we determine for all with .
For an -partite graph on vertex classes , we use to denote the family of -element subsets of that form -cliques in and for we use to denote the family of -element subsets in that contain . For any , we also use to denote
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank two anonymous referees for their helpful suggestions. The second author was supported by the National Natural Science Foundation of China (No. 11701407) and Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 183090222-S).
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