1. Introduction Given k 2, a k-uniform hypergraph (in short, k-graph) consists of a vertex set V and an edge set E ( V

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1 MINIMUM VERTEX DEGREE THRESHOLD FOR C 4-TILING JIE HAN AND YI ZHAO Abstract. We prove that the vertex degree threshold for tiling C4 (the - uniform hypergraph with four vertices and two triples in a -uniform hypergraph on n 4N vertices is ( n ( 4 n + n + c, where c = if n 8N and 8 c = otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling.. Introduction Given k, a k-uniform hypergraph (in short, k-graph consists of a vertex set V and an edge set E ( V k, where every edge is a k-element subset of V. Given a k-graph H with a set S of d vertices (where d k we define deg H (S to be the number of edges containing S (the subscript H is omitted if it is clear from the context. The minimum d-degree δ d (H of H is the minimum of deg H (S over all d-vertex sets S in H. Given a k-graph G of order g and a k-graph H of order n, a G-tiling (or G- packing of H is a subgraph of H that consists of vertex-disjoint copies of G. When g divides n, a perfect G-tiling (or a G-factor of H is a G-tiling of H consisting of n/g copies of G. Define t d (n, G to be the smallest integer t such that every k-graph H of order n gn with δ d (H t contains a perfect G-tiling. As a natural extension of the matching problem, tiling has been an active area in the past two decades (see surveys [5, ]. Much work has been done on the problem for graphs (k =, see e.g., [7,,, 6]. In particular, Kühn and Osthus [6] determined t (n, G, for any graph G, up to an additive constant. Tiling problems become much harder for hypergraphs. For example, despite much recent progress [, 5, 0,, 7, 4, 6], we still do not know the -degree threshold for a perfect matching in k-graphs for arbitrary k. Other than the matching problem, only a few tiling thresholds are known. Let K4 be the complete -graph on four vertices, and let K4 e be the (unique -graph on four vertices with three edges. Recently Lo and Markström [8] proved that t (n, K4 = (+o(n/4, and independently Keevash and Mycroft [9] determined the exact value of t (n, K4 for sufficiently large n. In [9], Lo and Markström proved that t (n, K4 e = ( + o(n/. Let C4 be the unique -graph on four vertices with two edges. This -graph was denoted by K4 e in [4], and by Y in [8]. Here we follow the notation in [4] and view it as a cycle on four vertices. Kühn and Osthus [4] showed that t (n, C4 = ( + o(n/4, and Czygrinow, DeBiasio Date: August, Mathematics Subject Classification. Primary 05C70, 05C65. Key words and phrases. graph packing, hypergraph, absorbing method, regularity lemma. The second author is partially supported by NSA grant H and NSF grant DMS

2 JIE HAN AND YI ZHAO and Nagle [4] recently determined t (n, C4 exactly for large n. In this paper we determine t (n, C4 for sufficiently large n. From now on, we simply write C4 as C. Previously we only knew t (n, K [, 7] and t (n, K4 4 [0] exactly, and t (n, K5 5 [], t (n, K(m and t (n, K4(m 4 [8] asymptotically, where Kk k denotes a single k-edge, and Kk k (m denotes the complete k-partite k-graph with m vertices in each part. So Theorem. below is one of the first (exact results on vertex degree conditions for hypergraph tiling. Theorem. (Main Result. Suppose H is a -graph on n vertices such that n 4N is sufficiently large and ( n δ (H 4 n + n + c(n, (. 8 where c(n = if n 8N and c(n = / otherwise. Then H contains a perfect C-tiling. Proposition. below shows that Theorem. is best possible. Theorem. and Proposition. together imply that t (n, C = ( n 4 n + 8 n + c(n. Proposition.. For every n 4N there exists a -graph of order n with minimum vertex degree ( n 4 n + 8n+c(n, which does not contain a perfect C-tiling. Proof. We give two constructions similar to those in [4]. Let V = A B with A = n 4 and B = n 4 +. A Steiner system S(,, m is a -graph S on n vertices such that every pair of vertices has degree one so S(,, m contains no copy of C. It is well-known that an S(,, m exists if and only if m, mod 6. Let H 0 = (V, E 0 be the -graph on n 8N vertices as follows. Let E 0 be the set of all triples intersecting A plus a Steiner system S(,, 4n + in B. Since for the Steiner system S(,, 4 n +, each vertex is in exactly 4 n/ = 8n edges, we have δ (H 0 = ( n n 4 + 8n. Furthermore, since B contains no copy of C, the size of the largest C-tiling in H 0 is A = n 4. So H 0 does not contain a perfect C-tiling. On the other hand, let H = (V, E be the -graph on n 4N \ 8N vertices as follows. Let G be a Steiner system of order 4 n+4. This is possible since 4 n+4 mod 6. Then pick an edge abc in G and let G be the induced subgraph of G on V (G\{a, b, c}. Finally let E be the set of all triples intersecting A plus G induced on B. Since G is a regular graph with vertex degree ( 4 n + 4 = 8 n +, we have that δ (G = 8 n + = 8 n. Thus, δ (H = ( n n n. As in the previous case, H does not contain a perfect C-tiling. As a typical approach of obtaining exact results, we distinguish the extremal case from the nonextremal case and solve them separately. Given a -graph H of order n, we say that H is C-free if H contains no copy of C. In this case, clearly, every pair of vertices has degree at most one. Every vertex has degree at most n because its link graph contains no vertex of degree two. Definition.. Given ɛ > 0, a -graph H on n vertices is called ɛ-extremal if there is a set S V (H, such that S ( ɛ n 4 and H[S] is C-free. Throughout the paper, we write A B for A B when sets A, B are disjoint. Given -graph H = (V, E and x V, the link graph of x has vertex set V \ {x} and the edge set {e \ {x} : e E(H, x e}.

3 MINIMUM VERTEX DEGREE THRESHOLD FOR C 4 -TILING Theorem.4 (Extremal Case. There exists ɛ > 0 such that for every -graph H on n vertices, where n 4N is sufficiently large, if H is ɛ-extremal and satisfies (., then H contains a perfect C-tiling. Theorem.5 (Nonextremal Case. For any ɛ > 0, there exists γ > 0 such that the following holds. Let H be a -graph on n vertices, where n 4N is sufficiently large. If H is not ɛ-extremal and satisfies δ (H ( 7 6 γ ( n, then H contains a perfect C-tiling. Theorem. follows Theorems.4 and.5 immediately by choosing ɛ from Theorem.4. The proof of Theorem.4 is somewhat routine and will be presented in in Section 4. The proof of Theorem.5, as the one of [4, Theorem.5], uses the absorbing method initiated by Rödl, Ruciński and Szemerédi, e.g., [, ]. More precisely, we find the perfect C-tiling by applying the Absorbing Lemma below and the C-tiling Lemma [8, Lemma.5] together. Lemma.6 (Absorbing Lemma. For any 0 < θ 0 4, there exist β > 0 and integer n.6 such that the following holds. Let H be a -graph of order n n.6 with δ (H ( 4 + θ( n. Then there is a vertex set W V (H with W 4N and W 049θn such that for any vertex subset U with U W =, U 4N and U βn both H[W ] and H[W U] contain C-factors. Lemma.7 (C-tiling Lemma, [8]. For any 0 < γ <, there exists an integer n.7 such that the following holds. Suppose H is a -graph on n > n.7 vertices with 7 n δ (H 6 γ, then H contains a C-tiling covering all but at most 9 /γ vertices or H is γ- extremal. We postpone the proof of Lemma.6 to Section and prove Theorem.5 now. Proof of Theorem.5. Without loss of generality, assume 0 < ɛ <. Let γ = ɛ and θ = 0 4 γ (thus θ < 0 4. We find β by applying Lemma.6. Choose n 4N such that n > max{n.6, n.7, 8 /(γβ}. Let H = (V, E be a -graph on n vertices. Suppose that H is not ɛ-extremal and δ (H ( 7 6 γ ( n. First we apply Lemma.6 to H and find the absorbing set W with W 049θn. Let H = H[V \ W ] and n = n W. Note that W < 0 4 θn = γn and thus n > n γn/ > n.7. Furthermore, δ (H δ (H W (n ( 7 n 7 6 γ 6 γ ( n Second we apply Lemma.7 to H with parameter γ in place of γ and derive that either H is γ-extremal or H contains a C-tiling covering all but at most 8 /γ vertices. In the former case, since ( γ n 4 > ( γ ( n γn > ( γ n 4 4 = ( ɛn 4, H is ɛ-extremal, a contradiction. In the latter case, let U be the set of uncovered vertices in H. Then we have U 4N and U 8 /γ βn by the choice of n. By Lemma.6, H[W U] contains a perfect C-tiling. Together with the C-tiling provided by Lemma.7, this gives a perfect C-tiling of H..

4 4 JIE HAN AND YI ZHAO The Absorbing Lemma and C-tiling Lemma in [4] are not very difficult to prove because of the co-degree condition. In contrast, our corresponding lemmas are harder. Luckily we already proved Lemma.7 in [8] (as a key step for finding a loose Hamilton cycle in -graphs. In order to prove Lemma.6, we will use the Strong Regularity Lemma and an extension lemma from [], which is a corollary of the counting lemma. The rest of the paper is organized as follows. We introduce the Regularity Lemma in Section, prove Lemma.6 in Section, and finally prove Theorem.4 in Section 4.. Regularity Lemma for -graphs.. Regular complexes. Before we can state the regularity lemma, we first define a complex. A hypergraph H consists of a vertex set V (H and an edge set E(H, where every edge e E(H is a non-empty subset of V (H. A hypergraph H is a complex if whenever e E(H and e is a non-empty subset of e we have that e E(H. All the complexes considered in this paper have the property that every vertex forms an edge. For a positive integer k, a complex H is a k-complex if every edge of H consists of at most k vertices. The edges of size i are called i-edges of H. Given a k-complex H, for each i [k] we denote by H i the underlying i-graph of H: the vertices of H i are those of H and the edges of H i are the i-edges of H. Given s k, a (k, s-complex H is an s-partite k-complex, by which we mean that the vertex set of H can be partitioned into sets V,..., V s such that every edge of H is crossing, namely, meets each V i in at most one vertex. Given i, an i-partite i-graph H and an i-partite (i -graph G on the same vertex set, we write K i (G for the family of all crossing i-sets that form a copy of the complete (i -graph K (i i in G. We define the density of H with respect to G to be d(h G := K i(g E(H K i (G if K i (G > 0, and d(h G = 0 otherwise. More generally, if Q = (Q,..., Q r is a collection of r subhypergraphs of G, we define K i (Q := r j= K i(q j and d(h Q := K i(q E(H K i (Q if K i (Q > 0, and d(h Q = 0 otherwise. We say that H is (d, δ, r-regular with respect to G if every r-tuple Q with K i (Q > δ K i (G satisfies d(h Q d δ. Instead of (d, δ, -regularity we simply refer to (d, δ-regularity. Given a (, -complex H, we say that H is (d, d, δ, δ, r-regular if the following conditions hold: ( For every pair K of vertex classes, H [K] is (d, δ-regular with respect to H [K] unless e(h [K] = 0, where H i [K] is the restriction of H i to the union of all vertex classes in K. ( H is (d, δ, r-regular with respect to H unless e(h = 0.

5 MINIMUM VERTEX DEGREE THRESHOLD FOR C 4 -TILING 5.. Statement of the Regularity Lemma. In this section we state the version of the regularity lemma due to Rödl and Schacht [5] for -graphs, which is almost the same as the one given by Frankl and Rödl [6]. We need more notation. Suppose that V is a finite set of vertices and P ( is a partition of V into sets V,..., V t, which will be called clusters. Given any j [], we denote by Cross j = Cross j (P ( the set of all crossing j-subsets of V. For every set A [t] we write Cross A for all the crossing subsets of V that meet V i precisely when i A. Let P A be a partition of Cross A. We refer to the partition classes of P A as cells. Let P ( be the union of all P A with A = (so P ( is a partition of Cross. We call P = {P (, P ( } a family of partitions on V. Given P = {P (, P ( } and K = v i v j v k with v i V i, v j V j and v k V k, the polyad P (K is the -partite -graph on V i V j V k with edge set C(v i v j C(v i v k C(v j v k, where e.g., C(v i v j is the cell in P i,j that contains v i v j. We say that P (K is (d, δ-regular if all C(v i v j, C(v i v k, C(v j v k are (d, δ-regular with respect to their underlying sets. We let ˆP ( be the family of all P (K for K Cross. Now we are ready to state the regularity lemma for -graphs. Theorem. (Rödl and Schacht [5], Theorem 7. For all δ > 0, t 0 N and all functions r : N N and δ : N (0, ], there are d > 0 such that /d N and integers T, n 0 such that the following holds for all n n 0 that are divisible by T!. Let H be a -graph of order n. Then there exists a family of partitions P = {P (, P ( } of the vertex set V of H such that ( P ( = {V,..., V t } is a partition of V into t clusters of equal size, where t 0 t T, ( P ( is a partition of Cross into at most T cells, ( for every K Cross, P (K is (d, δ(t -regular, (4 K (P δ V, where the summation is over all P ˆP ( such that H is not (d, δ, r(t -regular with respect to P for any d > 0... The Reduced -graph and the Extension Lemma. Given t 0 N and δ > 0, we choose functions r : N N and δ : N (0, ] such that the output of Theorem. satisfies the following hierarchy: { } { n 0 r, δ δ, d, }, (. T where r = r(t and δ = δ(t. Let H be a -graph on V of order n n 0 such that T! divides n. Suppose that P = {P (, P ( } satisfies Properties ( (4 given in Theorem.. For any d > 0, the reduced -graph R = R(H, P, d is defined as the -graph whose vertices are clusters V,..., V t and three clusters V i, V j, V k form an edge of R if there is some polyad P on V i V j V k such that H is (d, δ, r-regular with respect to P for some d d. Fact.. Let R = R(H, P, d be the reduced -graph defined above. If V i V j V k E(R, then there exists a (, -complex H on V i V j V k such that H is a subhypergraph of H and H is (d, d, δ, δ, r-regular for some d d. Proof. Since V i V j V k E(R, there exists a polyad P on V i V j V k such that H is (d, δ, r-regular with respect to P for some d d. Let H = P and H = E(H K (P. By Theorem., H is a (d, d, δ, δ, r-regular (, -complex.

6 6 JIE HAN AND YI ZHAO The following lemma says that the reduced -graph almost inherits the minimum degree condition from H. Its proof is almost identical to the one of [, Lemma 4.], which gives the corresponding result on co-degree. We thus omit the proof. Lemma.. In addition to (., suppose that δ, /t 0 d θ µ <. Let H be a -graph of order n n 0 such that T! divides n and δ (H (µ + θ ( n. Then in the reduced -graph R = R(H, P, d, all but at most θt vertices v V (R satisfy deg R (v µ ( t. Suppose that H is a (, -complex with vertex classes V, V, V, and G is a (, -complex with vertex classes X, X, X. A subcomplex H of H is called a partition-respecting copy of G if H is isomorphic to G and for each i [] the vertices corresponding to those in X i lie within V i. We write G H for the number of (labeled partition-respecting copies of G in H. Roughly speaking, the Extension Lemma [, Lemma 5] says that if G is an induced subcomplex of G, and H is suitably regular, then almost all copies of G in H can be extended to a large number of copies of G in H. Below we only state it for (, -complexes. Lemma.4 (Extension Lemma []. Let r, b, b, m 0 be positive integers, where b < b, and let c, θ, d, d, δ, δ be positive constants such that /d N and /m 0 {/r, δ} c min{δ, d } δ θ, d, /b. Then the following holds for all integers m m 0. Suppose that G is a (, -complex on b vertices with vertex classes X, X, X and let G be an induced subcomplex of G on b vertices. Suppose also that H is a (d, d, δ, δ, r-regular (, -complex with vertex classes V, V, V, all of size m and e(h > 0. Then all but at most θ G H labeled partition-respecting copies of G in H are extendible to at least cm b b labeled partition-respecting copies of G in H.. Proof of Lemma.6 In this section we prove Lemma.6 by using the lemmas introduced in Section. We remark that the constant 4 in Lemma.6 is best possible because if H consists of two disjoint cliques of order n/ each, then δ (H is about 4( n and any 4-vertex set that intersects both cliques can not be absorbed. For α > 0, i N and two vertices u, v V, we say that u is (α, i-reachable to v if and only if there are at least αn 4i (4i -sets W such that both H[u W ] and H[v W ] contain C-factors. In this case, we call W a reachable set for u and v. Similar definitions for absorbing method can be found in [8, 9]. Suppose that /n 0 {/r, δ} c min {δ, /T, d } δ, /t 0 d θ 0 4, and n 0 4T!/θ. Let H be a -graph on n n 0 +T! vertices with δ (H ( 4 +θ( n. We will prove that almost all pairs of vertices of H are (β 0, -reachable to each other, where β 0 = c /(5T 7. Claim.. There are at most 4θn pairs u, v ( V such that u is not (β0, - reachable to v.

7 MINIMUM VERTEX DEGREE THRESHOLD FOR C 4 -TILING 7 Proof. Let n N such that n n < T! and T! divides n. Then n n 0 4T!/θ. As θ 0 4, we have n n. Let H be an induced subhypergraph of H on any n vertices. Since n 4T!/θ, we have ( n δ (H 4 + θ T!(n 4 + θ n. We apply Theorem. to H, and let P be the the family of partitions, with clusters V,..., V t. Let m = n /t be the size of each cluster. Define the reduced -graph R = R(H, P, d on these clusters as in Section.. Let I be the set of i [t] such that deg R (V i < ( 4 + θ 4 ( t and let VI = i I V i. By Lemma., we have I θt/4 and thus V I (θt/4 m = θn /4. Let N(i be the set of vertices V j V (R \ {V i } such that {V i, V j } e for some e R. For any i [t] \ I, ( 4 + θ t N(i deg 4 R (V i implies that N(i ( + θ 8 t. Thus N(i N(j θ 4t for any i, j [t] \ I. Fix two not necessarily distinct i, j / I and V k N(i N(j. We pick V i and V j such that V i V k V i, V j V k V j R. Note that it is possible to have i = j or i = j or j = i. Let H be the (d, d, δ, δ, r-regular (, -complex with vertex classes V i, V k, V i provided by Fact., where d d. Let G be the (,-complex on X = {x, u}, X = {y, v}, X = {w} such that G = {xvw, uyw, uvw} and G is the family of all -subsets of the members of G. Note that in G both {x, u, v, w} and {y, u, v, w} span copies of C. Let G be the induced subcomplex of G on {u, v}. Since H, the highest level of the complex H, is not empty, by Lemma.4, all but at most θm ordered pairs (v i, v k V i V k are extendible to at least cm labeled copies of G in H, which implies that v i is (cm n, -reachable to v k. By averaging, all but at most θm vertices v i V i are (cm n, -reachable to at least m vertices of V k. We apply the same argument on V j, V k, V j and obtain that for all but at most θm vertices v j V j, v j is (cm n, -reachable to at least m vertices of V k. Thus for those v i and v j, there are m vertices v k V k such that both v i and v j are (cm n, -reachable to v k. Fix such v i, v j, v k. There are at least cm m reachable -sets for v i and v k from (V i, V i, V k avoiding v j. Fix one such -set, the number of -sets from (V j, V j, V k intersecting its three vertices is at most m. So the number of reachable 7-sets for v i, v j is at least m (cm m (cm m > c 4 m7 c 4 ( n T 7 > c 5 n 7 T 7 = β 0n 7, which means that v i is (β 0, -reachable to v j, where the last inequality holds because ( n n 7 ( > 4 5. Note that this is true for all but at most θm m = 6θm pairs of vertices in (V i, V j. Since there are at most ( t + t choices for Vi and V j, V I θn /4, and T! θn /4, there are at most ( t 6θm + t + ( V I + T!(n θm (t + t + θ n (n 4θn pairs u, v in V (H such that u is not (β 0, -reachable to v. Recall that it is possible to have vj V i or v j V i (when j = i or j = i.

8 8 JIE HAN AND YI ZHAO Proof of Lemma.6. Let β = β 0 0. Let V be the set of vertices v V such that at least n 64 vertices are not (β 0, -reachable to v. By Claim., V 5θn. There are two steps in our proof. In the first step, we build an absorbing family F such that for any small portion of vertices in V (H \ V, we can absorb them using members of F. In the second step, we put the vertices in V not covered by any member of F into a set A of copies of C. Thus, the union of F and A gives the desired absorbing set. We say that a set A absorbs another set B if A B = and both H[A] and H[A B] contains C-factors. Fix any 4-set S = {v, v, v, v 4 } V \ V, we will show that there are many 4-sets absorbing S. First, we find vertices u, u, u 4 such that v u u u 4 spans a copy of C, u i is (β 0, -reachable to v i, for i =,, 4. For the first condition, consider the link graph H v of v, which contains at least ( 4 + θ( n edges. By convexity, the number of paths of length two in Hv is ( deghv (x ( n x V \{v (n deg } H v (x x V \{v } ( ( (n 4 + θn > n, where the last inequality holds because θn. Since v u u u 4 spans a copy of C if u u u 4 is a path of length two in H v, then there are at least n choices for such u u u 4. Moreover, the number of triples violating the second condition is at most n (n 64 < 8 n. Thus, there are at least 8 n such u u u 4 satisfying both of the conditions. Second, we find reachable 7-sets C i for u i and v i, for i =,, 4, which is guaranteed by the second condition above. Since in each step we need to avoid at most previously selected vertices, there are at least β0 n7 choices for each C i. In total, we get 8 n ( β0 n7 > β0n sets F = C C C {u, u, u 4 } (because β 0 < c < 0 8. It is easy to see that F absorbs S. Indeed, H[F ] has a C-factor since C i {u i } spans two copies of C for i =,, 4. In addition, H[F S] has a C-factor since v u u u 4 spans a copy of C and C i {v i } spans two copies of C for i =,, 4. Now we choose a family F ( V 4 of 4-sets by selecting each 4-set randomly and independently with probability p = β 5 0n. Then F follows the binomial distribution B( ( ( n 4, p with expectation E( F = p n 4. Furthermore, for every 4-set S, let f(s denote the number of members of F that absorb S. Then f(s follows the binomial distribution B(N, p with N β0n 4 4 by previous calculation. Hence E(f(S pβ0n 4 4. Finally, since there are at most ( ( n 4 4 n < n47 pairs of intersecting 4-sets, the expected number of the intersecting pairs of 4-sets in F is at most p n47 = β0 0 n/. Applying Chernoff s bound on the first two properties and Markov s bound on the last one, we know that, with positive probability, F satisfies the following properties: F p ( n 4 < β 5 0 n, for any 4-set S, f(s p β4 0n 4 = β 9 0n/, the number of intersecting pairs of elements in F is at most β 0 0 n.

9 MINIMUM VERTEX DEGREE THRESHOLD FOR C 4 -TILING 9 Thus, by deleting one member from each intersecting pair and the non-absorbing members from F, we obtain a family F consisting of at most β0n 5 4-sets and for each 4-set S, at least β0n/ 9 β0 0 n > β0 0 n = βn members in F absorb S. At last, we will greedily build A, a collection of copies of C to cover the vertices in V not already covered by any member of F. Indeed, assume that we have built a < V 5θn copies of C. Together with the vertices in F, there are at most 4a + 4β0n 5 < 049θn vertices already selected. Then at most 049θn pairs of vertices intersect these vertices. So for any remaining vertex v V, there are at least n deg(v 05θn 4 + θ 049θn > n/ edges containing v and not intersecting the existing vertices, where the last inequality follows from θ 0 4. So there is a path of length two in the link graph of v not intersecting the existing vertices, which gives a copy of C containing v. Combining the vertices covered by A and F together, we get the desired absorbing set W satisfying W 4 5θn + 4β0n 5 < 049θn. 4. Proof of Theorem.4 In this section we prove Theorem.4. Our proof is similar to the one of [4, Theorem.4]. First let us start with some notation. Fix a -graph H = (V, E. Recall that the link graph of a vertex v V is a -graph on V \ {v}. Then for a set E of pairs in ( V (which can be viewed as a -graph, let degh (v, E = N H (v E. When E = X for some X V, we write degh (v, X as degh (v, X for short. Let deg H (v, E = E V \{v} degh (v, E. Given not necessarily disjoint subsets X, Y, Z of V, define e H (XY Z = {xyz E(H : x X, y Y, z Z} { ( V e H (XY Z = xyz \ E(H : x X, y Y, z Z We often omit the subscript H if it is clear from the context. The following fact is the only place where we need the exact degree condition (.. Fact 4.. Let H be a -graph on n vertices with n 4N satisfying (.. S V (H spans no copy of C, then S 4 n. Proof. Assume to the contrary, that S V (H spans no copy of C and is of size at least 4 n +. Take S 0 S with size exactly 4 n +. Then for any v S 0, deg(v, S 0 S0 = 8n. We split into two cases. Case. n 8N. In this case, for any v S 0, since deg(v, S 0 8n, we have that ( V S0 deg(v = deg(v, S 0 + deg v, \ ( n 8 n + 4 n < δ (H, contradicting (.. Case. n 4N \ 8N. }. If

10 0 JIE HAN AND YI ZHAO In this case, for any v S 0, deg(v, S 0 8 n implies that deg(v, S 0 8 n because n 4N \ 8N. So we have e(s 0 = ( deg(v, S 0 8 n 4 n + = n 4 n v S 0 However, neither n 4 n+4 8 or n+4 4 is a multiple of. Thus v S 0 deg(v, S 0 < n 4 4, which implies that there exists v 0 S 0 such that deg(v 0, S 0 < 8 n. Consequently, ( V S0 deg(v 0 = deg(v 0, S 0 +deg v 0, \ < ( n 8 n + 4 n δ (H, contradicting (.. Proof of Theorem.4. Take ɛ = 0 8 and let n be sufficiently large. We write α = ɛ / = 0 6. Let H = (V, E be a -graph of order n satisfying (. which is ɛ-extremal, namely, there exists a set S V (H such that S ( ɛ n 4 and H[S] is C-free. Let C V be a maximum set for which H[C] is C-free. Define { } C A = x V \ C : deg(x, C ( α, (4. and B = V \ (A C. We first claim the following bounds of A, B, C. Claim 4.. A > n 4 ( 4α, B < α n and n 4 ( ɛ C n 4. Proof. The estimate on C follows from our hypothesis and Fact 4.. We now estimate B. For any v C, we have deg(v, C C, which gives deg(v, C ( C C. By (., deg(v ( 4 n 8 n +. Thus ( ( V C deg v, \ 4 n 8 n + C + C ( 4 n C because C 4 n = 4 n C + 4 n + C. The estimate on C gives 4 n C ɛ < ( + ɛ( C. Hence ( V C deg v, \ < 4 n C + ( ( + ɛ( C + C = 4 n C + ( + ɛ( C (4. 4 ɛn + ( + ɛ( C < ɛn ( C. (4. Consequently e(cc(a B < ( C C ɛn ( C = ɛn. Together with the definition of A and B, we have C C C ( A B ɛn < e(cc(a B ( α B + A, 8

11 MINIMUM VERTEX DEGREE THRESHOLD FOR C 4-TILING so that A B ɛn < A + B α B. Since A and B are disjoint, we get that B < α n. Finally, A = n B C > n α n 4 n = n 4 ( 4α. In the rest of the section, we will build four vertex-disjoint C-tilings Q, R, S, T whose union is a perfect C-tiling of H. In particular, when A = n/4, B = and C = n/4, we have Q = R = S = and the perfect C-tiling T of H will be provided by Lemma 4.4. The purpose of C-tilings Q, R, S is covering the vertices of B and adjusting the sizes of A and C such that we can apply Lemma 4.4 after Q, R, S are removed. The C-tiling Q. Let Q be a largest C-tiling in H on B C and q = Q. We claim that B /4 q B. Since C contains no copy of C, every element of Q contains at least one vertex of B and consequently q B. On the other hand, suppose that q < B /4, then (B C \ V (Q spans no copy of C and has order B + C 4q > B + C B = C. which contradicts the assumption that C is a maximum C-free subset of V (H. Claim 4.. q + A n 4. Proof. Let l = n 4 A. There is nothing to show if l 0. If l =, we have B C = 4n +, and thus Fact 4. implies that H[B C] contains a copy of C. Thus q = l and we are done. We thus assume l and l > q B /4, which implies that B 4(l. In this case B because C 4 n. For any v C, by (4., we have deg(v, BC < ( 4 n C + ( + ɛ( C. By definition, 4 n C = A + B n 4 = B l. So we get e(bcc < C C 4 n C + ( + ɛ( C = ( + ɛ( B l +. Together with B 4(l, this implies C e(bcc > ( B ( + ɛ( B l + C = (( + ɛ(l ɛ B C C (( + ɛ(l 4ɛ(l = ( ɛ(l. (4.4 On the other hand, we want to bound e(bcc from above and then derive a contradiction. Assume that Q is the maximum C-tiling of size q such that each element of Q contains exactly one vertex in B and three vertices in C. Note that q because C is a maximum C-free set and B. Write B Q for the set of vertices of B covered by Q and C Q for the set of vertices of C covered by Q. Clearly, B Q = q, C Q = q and q q l. For any vertex v B \ B Q, deg(v, C q ( C + C < 4q C. Together with the definition of B and Claim 4., we get e(bcc = e(b Q CC + e((b \ B Q CC C C q ( α + B 4q C < q ( α + 4α nq C. (4.5

12 JIE HAN AND YI ZHAO Putting (4.4 and (4.5 together and using q l and C > n/, we get α = ɛ < α + 8α n C < α + 6α < α, which is a contradiction since α = 0 6. Let B and C be the vertices in B and C not covered by Q, respectively. By Claim 4., C C q C B > 4 n( ɛ α n > 4 n 4α n +. (4.6 The C-tiling R. Next we will build our C-tiling R which covers B such that every element in R contains one vertex from A, one vertex from B and two vertices from C. Since Q is a maximum C-tiling on B C, for every vertex v B, we have that. Together with (4.6, this implies that C deg(v, C C = C ( C deg(v, C C > ( 4 n 4α n. Together with (., we get that for every v B, ( deg(v, AC < 4 n 8 n + ( 4 n 4α n = n 4α n 4α n 4 n + < α n. By Claim 4. and (4.6, we have that A C > ( 4α n 4 ( 4 4α n > 7 n. Thus, deg(v, AC < α n < 7α A C, equivalently, deg(v, AC > ( 7α A C. For every v B, we greedily pick a copy of C containing v by picking a path of length two with center in A and two ends in C from the link graph of v. Suppose we have found i < B copies of C, then for any remaining vertex v B, by Claim 4., the number of pairs not intersecting the existing vertices is at least deg(v, AC i ( A + C > ( 7α A C B C > A, which guarantees a path of length two centered at A, so a copy of C containing v. Now all vertices of B are covered by Q or R. Let A denote the set of vertices of A not covered by Q or R and define C similarly. By the definition of Q and R, we have A = A B and C = B + C 4q B. Define s = 4 ( A C. Then s = 4 ( A B B C + 4q + B = 4 (4 A n + 4q = q + A n 4. Thus s Z, and s 0 by Claim 4.. Since q B, by Claim 4., s = q + A n 4 B + A n 4 = 4 n C ɛn. (4.7 4 The definition of Q and R also implies that C \ C B and C C B > C α C = ( 6α C, (4.8 where the second inequality follows from B < α n < α C.

13 MINIMUM VERTEX DEGREE THRESHOLD FOR C 4-TILING The C-tiling S. Next we will build our C-tiling S of size s such that every element of S contains two vertices in A and two vertices in C. Note that for any vertex v A, by (4. and (4.8, ( C deg(v, C α α 6α C C < α. Suppose that we have found i < s copies of C of the desired type. We next select two vertices a, a in A and note that they have at least ( 4α ( C common neighbors in C. By (4.7 and(4.8, C C ( 4α s C ( 4α ɛn C C ( 5α > 0. So we can pick a common neighbor c c of a and a from unused vertices of C such that {a, a, c, c } spans a copy of C. Let A be the set of vertices of A not covered by Q, R, S and define C similarly. Then A = A s = ( C A and C = C s = ( C A, so C = A. Furthermore, by (4.7 and (4.8, we have C = C s ( 6α C ɛn > ( 6α C ɛ C > ( 7α C. Hence, for every vertex v A, C deg(v, C α ( α 7α C ( C < α Since C ( 7α C ( 7α ( ɛ 4n, by (4., we know that for any vertex v C, deg(v, A C < ɛn ( C < ɛ C = 6ɛ A C. The C-tiling T. Finally we use the following lemma to find a C-tiling T covering A and C such that every element of T contains one vertex in A and three vertices in C. Note that in [4], this was done by applying a general theorem of Pikhurko [0, Theorem ] (but impossible here because we do not have the co-degree condition. Lemma 4.4. Suppose that 0 < ρ 0 6 and n 4.4 is sufficiently large. Let H be a -graph on n n 4.4 vertices with V (H = X Z such that = X. Further, assume that for every vertex v X, deg(v, Z ρ and for every vertex v Z, deg(v, XZ ρ X. Then H contains a perfect C-tiling. Applying Lemma 4.4 with X = A, Z = C, ρ = α finishes the proof of Theorem.4. Proof of Lemma 4.4. Let us outline the proof first. Let X = {x,..., x X }. Our goal is to partition the vertices of Z into X triples {Q,..., Q X } such that for every i =,..., X, {x i } Q i spans a copy of C in this case we say Q i and x i are suitable for each other. From our assumptions, every x X is suitable for most triples of C, and most triples of C are suitable for most vertices of X. However, once we partition C into a particular set of triples {Q,..., Q X }, we can not guarantee that every vertex in X is suitable for most Q i s. To handle this difficulty, we use the absorbing method first find a small number of triples that can absorb any small(er amount of vertices of X and then extend it to a partition {Q,..., Q X } covering Z, and finally apply the greedy algorithm and the Marriage.

14 4 JIE HAN AND YI ZHAO Theorem to find a perfect matching between X and {Q,..., Q X }. Note that a similar approach was outlined in [0] to prove the extremal case. We now start our proof. Let G be the graph of all pairs uv in Z such that deg(uv, X ( ρ X. We claim that δ(g ρ. (4.9 Otherwise, some vertex v Z satisfies deg G (v < ρ, equivalently, deg G (v > ρ. As each u / N G (v satisfies deg(uv, X > ρ X, we have deg(v, XZ > ρ ρ X = ρ X, contradicting our assumption. We call a triple z z z in Z good if G[z z z ] contains at least two edges, otherwise bad. Since a bad triple contains at least two non-edges of G, by (4.9, the number of bad triples in Z is at most ( degg (x ρ ρ. x Z If z z z is good, then by the definition of G, it is suitable for at least ( ρ X vertices of X. On the other hand, for any x X, consider the link graph H x of x on Z, which contains at least ( ρ edges. By convexity, the number of triples z z z suitable for x is at least deghx (z ( ρ( > ( ρ, z Z where the last inequality holds because is large enough. Thus, the number of good triples z z z suitable for x is at least ( ρ ρ ( = ( 5ρ. Let F 0 be the set of good triples in Z. We want to form a family of disjoint good triples in Z such that for any x X, many triples from this family are suitable for x. To achieve this, we choose a subfamily F from F 0 by selecting each member randomly and independently with probability p = ρ /4. Then F follows the binomial distribution B( F 0, p with expectation E( F = p F 0 p. Furthermore, for every x X, let f(x denote the number of members of F that are suitable for x. Then f(x follows the binomial distribution B(N, p with N ( 5ρ ( by previous calculation. Hence E(f(x p( 5ρ. Finally, since there are at most ( < 4 5 pairs of intersecting triples, the expected number of the intersecting triples in F is at most p 4 5 = ρ /. By applying Chernoff s bound on the first two properties below and Markov s bound on the last one, we can find a family F F 0 that satisfies F p < ρ/4, for any vertex x X, at least p ( 5ρ > 6 ρ/4 ( 6ρ triples in F are suitable for x, the number of intersecting pairs of triples in F is at most ρ /. After deleting one triple from each of the intersecting pairs from F, we obtain a subfamily F consisting of at most ρ/4 disjoint good triples in Z and for each x X, at least ρ /4 6 ( 6ρ ρ/ > ρ/4 (4.0 members of F are suitable for x, where the inequality holds because ρ 0 6.

15 MINIMUM VERTEX DEGREE THRESHOLD FOR C 4-TILING 5 Denote F by {Q,..., Q q } for some q ρ/4. Let Z be the set of vertices of Z not in any element of F. Define G = G[Z ]. Note that Z = q. For every v Z, deg G (v deg G (v ρ by (4.9. Thus δ(g Z ρ = q ρ > Z, because ρ 0 6. By Dirac s Theorem, G is Hamiltonian. We thus find a Hamiltonian cycle of G, denoted by b b b m b, where m = X q. Let Q q+i = b i b i b i for i m. Then Q q+,..., Q X are good triples. Now consider the bipartite graph Γ between X and Q := {Q, Q,..., Q X }, such that x X and Q i Q are adjacent if and only if Q i is suitable for x. For every Q i, since it is good, deg Γ (Q i ( ρ X. Let Q = {Q q+,..., Q X }. Let X 0 be the set of x X such that deg Γ (x, Q Q /. Then X 0 Q e Γ (X, Q ρ X Q, which implies that X 0 4 ρ X = 4 ρ. We now find a perfect matching between X and Q as follows. Step. Each vertex x X 0 is matched to a different member of F that is suitable for x this is possible because of (4.0 and X 0 4 ρ ρ/4 since ρ 0 6. Step. Each of the unused triples in Q Q Q q is matched to a suitable vertex in X \ X 0 this is possible because deg Γ (Q i ( ρ X q. Step. Let X be the set of the remaining vertices in X. Then X = X q = Q. Now consider Γ = Γ[X, Q ]. It is easy to check that δ(γ X / thus Γ contains a perfect matching by the Marriage Theorem. The perfect matching between X and Q gives rise to the desired perfect C-tiling of H as outlined in the beginning of the proof. Acknowledgment We thank Richard Mycroft for helpful discussion on the Regularity Lemma and Extension Lemma. We also thank two anonymous referees for their valuable comments that improved the presentation. References [] N. Alon, P. Frankl, H. Huang, V. Rödl, A. Ruciński, and B. Sudakov. Large matchings in uniform hypergraphs and the conjecture of Erdős and Samuels. J. Combin. Theory Ser. A, 9(6:00 5, 0. [] N. Alon and R. Yuster. H-factors in dense graphs. J. Combin. Theory Ser. B, 66(:69 8, 996. [] O. Cooley, N. Fountoulakis, D. Kühn, and D. Osthus. Embeddings and Ramsey numbers of sparse k-uniform hypergraphs. Combinatorica, 9(:6 97, 009. [4] A. Czygrinow, L. DeBiasio, and B. Nagle. Tiling -uniform hypergraphs with K4 e. Journal of Graph Theory, 75(:4 6, 04. [5] A. Czygrinow and V. Kamat. Tight co-degree condition for perfect matchings in 4-graphs. Electron. J. Combin., 9(:Paper 0, 6, 0. [6] P. Frankl and V. Rödl. Extremal problems on set systems. Random Structures Algorithms, 0(: 64, 00. [7] A. Hajnal and E. Szemerédi. Proof of a conjecture of P. Erdős. In Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 969, pages North-Holland, Amsterdam, 970.

16 6 JIE HAN AND YI ZHAO [8] J. Han and Y. Zhao. Minimum degree thresholds for loose Hamilton cycle in -graphs. submitted. [9] P. Keevash and R. Mycroft. A geometric theory for hypergraph matching. Memoirs of the American Mathematical Society, (Monograph 908, 04. [0] I. Khan. Perfect matchings in 4-uniform hypergraphs. arxiv: [] I. Khan. Perfect matchings in -uniform hypergraphs with large vertex degree. SIAM J. Discrete Math., 7(:0 09, 0. [] J. Komlós, G. Sárközy, and E. Szemerédi. Proof of the Alon-Yuster conjecture. Discrete Math., 5(-:55 69, 00. Combinatorics (Prague, 998. [] D. Kühn, R. Mycroft, and D. Osthus. Hamilton l-cycles in uniform hypergraphs. Journal of Combinatorial Theory. Series A, 7(7:90 97, 00. [4] D. Kühn and D. Osthus. Loose Hamilton cycles in -uniform hypergraphs of high minimum degree. Journal of Combinatorial Theory. Series B, 96(6:767 8, 006. [5] D. Kühn and D. Osthus. Embedding large subgraphs into dense graphs. In Surveys in combinatorics 009, volume 65 of London Math. Soc. Lecture Note Ser., pages Cambridge Univ. Press, Cambridge, 009. [6] D. Kühn and D. Osthus. The minimum degree threshold for perfect graph packings. Combinatorica, 9(:65 07, 009. [7] D. Kühn, D. Osthus, and A. Treglown. Matchings in -uniform hypergraphs. J. Combin. Theory Ser. B, 0(:9 05, 0. [8] A. Lo and K. Markström. F -factors in hypergraphs via absorption. Graphs and Combinatorics, to appear. [9] A. Lo and K. Markström. Minimum codegree threshold for (K4 e-factors. J. Combin. Theory Ser. A, 0(:708 7, 0. [0] O. Pikhurko. Perfect matchings and K4 -tilings in hypergraphs of large codegree. Graphs Combin., 4(4:9 404, 008. [] V. Rödl and A. Ruciński. Dirac-type questions for hypergraphs a survey (or more problems for endre to solve. An Irregular Mind, Bolyai Soc. Math. Studies :56 590, 00. [] V. Rödl, A. Ruciński, and E. Szemerédi. A Dirac-type theorem for -uniform hypergraphs. Combinatorics, Probability and Computing, 5(-:9 5, 006. [] V. Rödl, A. Ruciński, and E. Szemerédi. An approximate Dirac-type theorem for k-uniform hypergraphs. Combinatorica, 8(:9 60, 008. [4] V. Rödl, A. Ruciński, and E. Szemerédi. Perfect matchings in large uniform hypergraphs with large minimum collective degree. J. Combin. Theory Ser. A, 6(:6 66, 009. [5] V. Rödl and M. Schacht. Regular partitions of hypergraphs: regularity lemmas. Combin. Probab. Comput., 6(6:8 885, 007. [6] A. Treglown and Y. Zhao. Exact minimum degree thresholds for perfect matchings in uniform hypergraphs II. J. Combin. Theory Ser. A, 0(7:46 48, 0. (Jie Han and Yi Zhao Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 00 address, Jie Han: jhan@gsu.edu address, Yi Zhao: yzhao6@gsu.edu