Turán problems for vertex-disjoint cliques in multi-partite hypergraphs

https://doi.org/10.1016/j.disc.2020.112005Get rights and content

Abstract

For two s-uniform hypergraphs H and F, the Turán number exs(H,F) is the maximum number of edges in an F-free subgraph of H. Let s,r,k,n1,,nr be integers satisfying 2sr and n1n2nr. De Silva, Heysse and Young determined ex2(Kn1,,nr,kK2) and De Silva, Heysse, Kapilow, Schenfisch and Young determined ex2(Kn1,,nr,kKr). In this paper, as a generalization of these results, we consider three Turán-type problems for k disjoint cliques in r-partite s-uniform hypergraphs. First, we consider a multi-partite version of the Erdős matching conjecture and determine exs(Kn1,,nr(s),kKs(s)) for n1s3k2+sr. Then, using a probabilistic argument, we determine exs(Kn1,,nr(s),kKr(s)) for all n1k. Recently, Alon and Shikhelman determined asymptotically, for all F, the generalized Turán number ex2(Kn,Ks,F), which is the maximum number of copies of Ks in an F-free graph on n vertices. Here we determine ex2(Kn1,,nr,Ks,kKr) with n1k and n3==nr. Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szabó, we determine ex2(Kn1,,nr,Ks,kKr) for all n1,,nr with n4rr(k1)k2r2.

Introduction

An s-uniform hypergraph, or simply an s-graph, is a hypergraph whose edges have exactly s vertices. For an s-graph H, let V(H) be the vertex set of H and E(H) the edge set of H. An s-graph H is called F-free if H does not contain any copy of F as a subgraph. For two s-graphs H and F, the Turán number exs(H,F) is the maximum number of edges of an F-free subgraph of H. Denote by Kt(s) the complete s-graph on t vertices. A copy of Kt(s) in an s-graph H is also called a t-clique of H. Let kKt(s) denote the s-graph consisting of k vertex-disjoint copies of Kt(s). If t=s, then kKs(s) represents a matching of size k. Let n1,,nr be integers and V1,V2,,Vr be disjoint vertex sets with |Vi|=ni for each i=1,,r. A complete r-partite s-graph on vertex classes V1,V2,,Vr, denoted by K(s)(V1,V2,,Vr) or Kn1,n2,,nr(s), is defined to be the s-graph whose edge set consists of all the s-element subsets S of V1V2Vr such that |SVi|1 for all i=1,,r. An s-graph H is called an r-partite s-graph on vertex classes V1,V2,,Vr if H is a subgraph of K(s)(V1,V2,,Vr). For s=2, we often write Kt,kKt,K(V1,V2,,Vr),Kn1,n2,,nr and ex(H,F) instead of Kt(2),kKt(2),K(2)(V1,V2,,Vr),Kn1,n2,,nr(2) and ex2(H,F). Let [n] denote the set {1,2,,n} and [m,n] denote the set {m,m+1,,n} for mn.

Turán-type problems were first considered by Mantel [15] in 1907, who determined ex(Kn,K3). In 1941, Turán [17] showed that the balanced complete t-partite graph on n vertices, called the Turán graph and denoted by Tn,t, is the unique graph that maximizes the number of edges among all Kt+1-free graphs on n 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 exs(Kn(s),Kt(s)) with t>s>2 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 ex(Kn1,,nr,kK2).

Theorem 1.1

For n1n2nr and kn1, ex(Kn1,n2,,nr,kK2)=(k1)(n2++nr).

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 ex(Kn1,,nr,kKr).

Theorem 1.2 [5]

For n1n2nr and kn1, ex(Kn1,,nr,kKr)=1i<jrninjn1n2+(k1)n2.

In this paper, we consider three Turán-type problems for k disjoint cliques in r-partite s-graphs. Let n1,n2,,nr be integers. For any A[r], denote iAni by nA. Define fk(s)(n2,,nr)=(k1)A:A[2,r]|A|=s1nA,gk(s)(n1,n2,,nr)=A:A[r]|A|=snAn[s]+(k1)n[2,s], and hk(s)(n1,n2,,nr)=A:A[r]|A|=s,{1,2}AnA+A:A[3,r]|A|=s2(k1)n2nA.

Theorem 1.3

For 2sr, k1 and n1n2nr, if n1s3k+sr for sr2; n1s3k2+sr for s=r1 and n1k for s=r, then exs(Kn1,n2,,nr(s),kKs(s))=fk(s)(n2,,nr).

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 exs(Kn(s),kKs(s))=maxks1s,nsnk+1sand is still open when n is close to s(k1), see [4], [6], [8], [9], [10] for recent progress. The lower bound in Theorem 1.3 follows from the following construction. Let H1 be an r-partite s-graph on vertex classes V1,V2,,Vr with sizes n1,n2,,nr, respectively. Let V1 be a (k1)-element subset of V1. An edge S of K(s)(V1,V2,,Vr) forms an edge of H1 if and only if SV1. It is easy to see that H1 is kKs(s)-free. Otherwise, if H1 has a matching of size k, then we have |V1|k since each edge of H1 contains a vertex in V1.

As our second main result, we use a probabilistic argument to determine exs(Kn1,,nr(s),kKr(s)).

Theorem 1.4

For 2sr, n1n2nr and kn1, exsKn1,,nr(s),kKr(s)=gk(s)(n1,n2,,nr).

The lower bound in Theorem 1.4 follows from the following construction. Let H2 be an r-partite s-graph on vertex classes V1,V2,,Vr with sizes n1,n2,,nr, respectively. Let V1 be an (n1k+1)-element subset of V1 and let H2 be obtained by deleting all the edges of K(s)(V1,V2,,Vs) from K(s)(V1,V2,,Vr). It is easy to see that H2 is kKr(s)-free. Otherwise, if there are k vertex-disjoint copies of Kr(s) in H2, then we have |V1V1|k since each copy of Kr(s) in H2 contains a vertex in V1V1.

We also consider the generalized Turán problem in multi-partite graphs. Let ex(G,T,F) denote the maximum number of copies of T in an F-free subgraph of G. The first result of this type is due to Zykov [18], who showed that the Turán graph also maximizes the number of s-cliques in an n-vertex Kt+1-free graph for st. Recently, Alon and Shikhelman [2] determined ex(Kn,Ks,F) asymptotically for any F with chromatic number χ(F)=t+1>s. Precisely, they proved that ex(Kn,Ks,F)=ks(Tn,t)+o(ns),where ks(Tn,t) denotes the number of s-cliques in the Turán graph Tn,t. 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 s-cliques in a kKr-free subgraph of Kn1,,nr. By the same probabilistic argument as in the proof of Theorem 1.4, we obtain the following result.

Theorem 1.5

For 2sr, n1n2n3 and kn1, ex(Kn1,n2,n3,,n3r2,Ks,kKr)=hk(s)(n1,n2,n3,,n3r2).

Note that for r=3, s3 and arbitrary n1,n2,n3, the Turán number exKn1,n2,n3,Ks,kK3 is determined by Theorem 1.5. Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szabó [11], we also determine ex(Kn1,,nr,Ks,kKr) for r4 and n4 sufficiently larger than k.

Theorem 1.6

For r4, 2sr, n1n2nr and kn1, if n4rr(k1)k2r2, then exKn1,,nr,Ks,kKr=hk(s)(n1,n2,,nr).

The lower bounds in Theorem 1.5, Theorem 1.6 follow from the same construction as follows. Let G be an r-partite graph on V1,V2,,Vr, which are of sizes n1,n2,,nr, respectively. Let V1 be an (n1k+1)-element subset of V1. Then G is obtained by deleting all the edges of K(V1,V2) from K(V1,V2,,Vr). It is easy to see that G is kKr-free. Otherwise, if there are k vertex-disjoint copies of Kr in G, then we have |V1V1|k since each copy of Kr in G contains a vertex in V1V1.

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 kKs(s) in r-partite s-graphs

In this section, we prove Theorem 1.3. First, we consider the case s=r, which is the base case for other results in this paper. Aharoni and Howard [1] determined the maximum number of edges in a balanced r-partite r-graph that is kKr(r)-free. By the same argument, we prove the following result:

Lemma 2.1

For any integers 1kn1n2nr, exr(Kn1,,nr(r),kKr(r))=(k1)n2nr.

Proof

We shall partition the edge set of K(r)(V1,,Vr) into n2n3nr matchings of size n1 each. Let Vi={vi,0,vi,1,,vi,ni1} for i=1,2,,r and Λ

Turán number of kKr(s) in r-partite s-graphs

In this section, we generalize the result of [5] to s-graphs by using a probabilistic argument. The following lemma will be useful for us.

Lemma 3.1

Assume that b>0, w1w2wN>0 and let (P) be a linear programming model as follows: maxz=i=1Nxis.t.i=1Nwi1xib,0xiwi,i=1,2,,N. Let M be the integral part of b and a=wM+1(bM). Then i=1Mwi+a is the optimal value of (P).

Proof

Suppose to the contrary that there exists a feasible solution y=(y1,y2,,yN) to (P) such that i=1Nyi>i=1Mwi+a.Since y is a feasible

The number of s-cliques in r-partite graphs

In this section, we first determine ex(Kn1,,nr,Ks,kKr) for the case n1n2n3=n4==nr. Then, by utilizing a result on rainbow matchings, we determine ex(Kn1,,nr,Ks,kKr) for all n1,,nr with n4rr(k1)k2r2.

For an r-partite graph G on vertex classes V1,V2,,Vr, we use Ks(G) to denote the family of s-element subsets of V(G) that form s-cliques in G and for uV(G) we use Ks(u,G) to denote the family of s-element subsets in Ks(G) that contain u. For any A[r], we also use Ks(VA) to denote Ks(G[VA])

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).

References (18)

There are more references available in the full text version of this article.

Cited by (2)

View full text