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1 Technical Report TR Short Paths in Quasi-random Triple Systems With Sparse Underlying Graphs by Joanna Polcyn, Vojtech Rodl, Andrzej Rucinski, Endre Szemeredi Mathematics and Computer Science EMORY UNIVERSITY

2 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI, AND ENDRE SZEMERÉDI ABSTRACT. The Frankl-Rödl regularity lemma for 3-uniform hypergraphs asserts that every large hypergraph can be decomposed into a bounded number of quasi-random structures consisting of a subhypergraph and a sparse underlying graph. In this paper we show that in such a quasi-random structure most pairs of the edges of the graph can be connected by hyperpaths of length at most twelve. Two applications are also given.. INTRODUCTION The Regularity Lemma from [9] is a powerful tool in contemporary graph theory and combinatorics. It allows one to partition every large graph into a bounded number of bipartite subgraphs, most of which are quasi-random, that is, they possess essentially all typical properties of corresponding random graphs. One of these properties, quite easy to prove, is that every two vertices with non-negligible neighborhoods can be connected by a path of length at most four (see, e.g., [6] and Corollary 8.5(a) in the Appendix below). In this paper we study the much harder problem of the existence of short paths in 3- uniform, 3-partite hypergraphs with a certain regular structure related to the Frankl-Rödl regularity lemma in []. When this lemma is being applied, the initial hypergraph is broken into several quasi-random pieces and a desired structure is built from segments scattered among these highly regular substructures. It is then important to sew them together by relatively short hyperpaths. Two examples of this approach can be found in the forthcoming papers [8] and [2], where, respectively, the existence of Hamilton cycles in 3-uniform hypergraphs and the Ramsey numbers for hypercycles are treated. In both these applications, besides the Frankl- Rödl Lemma itself, a crucial role is played by the Connection Lemma, analogous to, but much more complicated than the above mentioned result for graphs. The goal of this paper is to prove this Connection Lemma for quasi-random, 3-uniform hypergraphs. In the next section, after some preliminary definitions, we state our main result, Theorem 2.9. Then, in section 3 we reformulate it in a more constructive way, specifying, in terms of their fourth neighborhoods, the edges that can be connected by short hyperpaths. Section 4 contains proofs of our main results, both relying on two lemmas, Lemma 4. and Lemma 4.2, which themselves will be proved in Sections 5 and 6. Section 7 presents briefly two applications of Theorem 2.9. One of them, a blow-up type result, guarantees a subhamiltonian path in a quasi-random 3-uniform hypergraph. The other approximates every large Date: September 8, 2004.

3 2 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS 3-uniform hypergraph by finitely many pieces of small diamater. Finally, the Appendix collects elementary facts aboutɛ-regular graphs which are used throughout the paper. Acknowledgements. The authors would like to extend their deepest gratitude to Brendan Nagle for his invaluable contribution to the initial version of this paper. Without his persistence in comprehending our elusive ideas this work would have never be born. 2. PRELIMINARIES AND MAIN RESULT Definition 2.. A 3-uniform hypergraph is a pairh = (V, E), where V is a finite set of vertices and E ( V 3) is a family of 3-element subsets of V called hyperedges or triplets. Throughout the paper we will often identifyh with E. We callh 3-partite if there exists a partition V= V V 2 V 3 such that for each e E and for each i=, 2, 3 we have e V i. We refer to any 3-partite 3-uniform hypergraph H with a fixed 3-partition (V, V 2, V 3 ) as a 3-graph. For an arbitrary hypergraphh and a graph G on the same vertex set, we denote byh G the subhypergraph ofh obtained by removing all hyperedges containing at least one edge of G. The density andɛ-regularity of bipartite graphs is measured by the ratio of edges to all potential edges (see the Appendix). For 3-graphs it is the ratio of hyperedges coinciding with the triangles of an underlying graph to all triangles in that graph. Definition 2.2. For a 3-partite graph P with a fixed 3-partition V V 2 V 3, we shall write P=P 2 P 23 P 3, where P i j ={xy P: x V i, y V j }. Furthermore, let Tr(P) be the set of all (vertex sets of) triangles formed by the edges of P. If P=P 2 P 23 P 3 is a 3-partite graph with the same vertex partition ash, and moreover,h Tr(P), then we say that P underliesh. The natural notion of density ofh with respect to P counts the proportion of triangles of P which are triplets ofh, and then theδ-regularity ofh means that for all Q P that contain a δ-fraction of T r(p), the densities of H with respect to such Q s are within δ from each other. However, it turns out that in some applications this is not strong enough. Therefore, the concept of so called (δ, r)-regularity was introduced in []. Definition 2.3. Let r be an integer and leth be a 3-graph with an underlying 3-partite graph P=P 2 P 23 P 3. LetQ=(Q(),..., Q(r)) be an r-tuple of 3-partite subgraphs Q(s)=Q 2 (s) Q 23 (s) Q 3 (s) satisfying that for all s {, 2,..., r} and i< j 3, Q i j (s) P i j. We define the density d H (Q) ofh with respect toqas () d H (Q)= H r s= Tr(Q(s)) r, s= Tr(Q(s)) if r s= T r(q(s)) > 0, and 0 otherwise. Definition 2.4. Let an integer r and real numbers 0<α,δ< be given. We say that a 3-graphH is (α,δ, r)-regular with respect to an underlying graph P=P 2 P 23 P 3 if

4 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 3 for any r-tuple of subgraphsq=(q(),..., Q(r)) as above, if r Tr(Q(s)) >δ Tr(P), then (2) d H (Q) α <δ. s= We say thath is (δ, r)-regular with respect to P if it is (α,δ, r)-regular for someα. Note that ifh is (δ, r)-regular with respect to P,δ δ, and r r is an integer, thenh is also (δ, r )-regular with respect to P (with the sameα). If r=, we just use the names δ-regular and (α,δ)-regular. Setup 2.5. In what follows we always assume thath is a 3-graph and P=P 2 P 23 P 3 is a 3-partite graph, both with the same 3-partition V= V(H)=V(P)=V V 2 V 3 with V = V 2 = V 3 =n, and moreover, that P underliesh, i.e.,h Tr(P). Definition 2.6. Given H and P as in Setup 2.5, integers l and r and real numbers α, δ and ɛ, we call the pair (H, P) an (α,δ, l, r,ɛ)-triad if (i) each P i j, i< j 3, is (/l,ɛ)-regular; (ii) H is (α, δ, r)-regular with respect to P. In particular, it follows that if (H, P) is an (α,δ, l, r,ɛ)-triad then for all i< j 3 we have (3) (/l ɛ)n 2 < P i j <(/l+ɛ)n 2 The hypergraph regularity lemma in [] states that with the right choice of parameters, for every large 3-uniform hypergraphh= (V, E) the complete graph on V can be partitioned into finitely many graphs so that most triplets ofh belong to (α,δ, l, r,ɛ)-triad built upon these graphs. This paper studies the structure of (H, P) in such a typical situation. There are several ways to define a path in a 3-uniform hypergraph, and we choose one in which the edges are glued along the path in the most tight way (see [5] and [3] for some study of paths and cycles defined in a loose way). Definition 2.7. LetH be a 3-uniform hypergraph. A hyperpath of length k 0 inh is a subhypergraphpofh consisting of k+2 vertices and k hyperedges and whose vertices can be labelled x,..., x k+2 so that for each i=,...,k, x i x i+ x i+2 H. We then say thatp goes from the pair x x 2 to the pair x k+2 x k+ and these two pairs are called the endpairs of P. The vertices x 3,..., x k are called internal. Two paths are said to be internally disjoint if they do not share any internal vertex. Remark 2.8. Note that the endpairs are ordered pairs of vertices. IfH is a 3-partite hypergraph then the vertices of any hyperpath traverse the partition sets only in the cyclic order V V 2 V 3 V, or in its reverse (see Figure ). Hence, there are pairs of ordered pairs of vertices which, even in a complete 3-graph, are not connected by any hyperpath. Another consequence is that the lengths of paths connecting two given endpairs are equal modulo 3.

5 4 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS Throughout the paper we will be assuming that the cyclic ordering V V 2 V 3 V is canonical, and thus, specifying two unordered pairs of vertices, e and f, and saying that a hyperpath goes from e to f will not be ambiguous. (Note that under this convention a hyperpath from f to e is not a mere reverse of a path from e to f.) Note also that unlike the graph case, the length of the shortest hyperpath between two given endpairs does not satisfy the triangle inequality, and thus cannot be called distance. Our goal is to prove the following Connection Lemma which, in a way, extends a simple fact about graphs, Corollary 8.5(b) (see Appendix), to 3-uniform quasi-random hypergraphs. In addition, for the sake of future applications, we may force the hyperpaths to avoid a specified set of vertices S. A hyperpathpis called S -avoiding if V(P) S =. Not to face the burden of computing yet another constant, we restrict S to have size only at most n/ log n. (The numerical constants are, clearly, not best possible.) Theorem 2.9 (Connection Lemma). For all realα (0, ) and for allδ<δ 0, where α 49 δ 0 = , there exist two sequences r(l) andɛ(l) so that for allh, P and for integer l if (H, P) is an (α,δ, l, r(l),ɛ(l))-triad with V = V 2 = V 3 = n sufficiently large, then there is a subgraph P 0 of at most 27 δn 2 /l edges of P such that for every ordered pair of disjoint edges (e, f ) (P P 0 ) (P P 0 ), e f=, and for every set S V(H)\(e f ) of size S n/ log n, there is inh P 0 an S -avoiding hyperpath from e to f of length at most twelve. Remark 2.0. In principle it might happen that an edge e P P 0 is isolated inh P 0, that is, all triplets containing e also contain an edge of P 0. The conclusion of the above theorem ensures that this is not the case. In fact, all edges e P P 0 are mutually connected by short hyperpaths withinh P 0. V k V j V i e f FIGURE. A hyperpath of length 2 from e to f. Every 3 consecutive vertices on the path form a hyperedge. 3. CONSTRUCTIVE REFORMULATION As mentioned earlier, in the case of (d,ɛ)-regular graphs, it is easy to see that for every pair of vertices with at leastɛn neighbors each, there is a short path (of length at most four)

6 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 5 between them (see, e.g., [6] and the Appendix below). In fact, see [7], every two vertices of degree at least 6(ɛ 2 /d)n can be connected by a path of length at most five. The quantification of Theorem (note that there exist sequences r(l) and ɛ(l) translates to for all l there exist r andɛ ) indicates the possibility of the following hierarchy of constants: α δ /l /r,ɛ, whereβ γ means thatγis sufficiently smaller thanβ, or thatγis chosen only afterβis being fixed. Polcyn [6], working under a comfortable assumption that δ /l, proved that most edges of P can be mutually connected by hyperpaths of length at most seven. Typical edges were defined in [6] in terms of the first and second neighborhood inh. Here, withδ and /l swapped in the hierarchy, to formulate a constructive version of Theorem 2.9, we need to look into the fourth neighborhood of an edge. Let us begin by defining the first neighborhood. Definition 3.. LetH be a 3-uniform hypergraph and let e={x, y} be a pair of vertices in V= V(H). We define the hypergraph neighborhood of e to beγ H (e)={z V :{z, x, y} H}. The vertices inγ H (e) will be called neighbors of e. Note that in a 3-graphH with an underlying graph P=P 2 P 23 P 3, if e P i j then Γ H (e) V k, where{i, j, k}={, 2, 3}. Imagine that both,h and P are chosen at random as a result of the following 2-round experiment. First, create P by tossing a coin over each pair in (V V 2 ) (V 2 V 3 ) (V V 3 ) independently with the success probability /l, then createh by selecting each triangle of P with probabilityα. In such a random hypergraph the expected number of triplets isαn 3 /l 3 and, for a given edge of P (here we condition that e has been selected), the expected value of Γ H (e) equalsαn/l 2. It is proved in [6] that if (H, P) is an (deterministic) (α,δ, l,,ɛ(l))- triad, then for almost all edges of P, Γ H (e) is close to the above expectation. Fact 3.2 ([6]). For all realα>0 andδ>0, there exists a sequenceɛ(l)>0 such that for all integer l, whenever (H, P) is an (α,δ, l,,ɛ(l))-triad then all but at most 7 δn 2 /l edges of P i j, i< j 3, satisfy the inequalities n ( l ɛ ) 2 (α δ)< Γ H (e) <(α+δ) ( ) 2 l +ɛ n. Remark 3.3. In [6], the above inequalities contain only the term 4 δ in place ofδ, but the same proof yields also Fact 3.2 in the present form. (The constant 7 instead of 6 in [6] comes from considering here all edges of P i j, and not just the proper ones.) However, for reasons which will be explained later, to guarantee short connections (via hyperpaths) of an edge e P with most of the other edges ofh, we will need to look four steps ahead. Definition 3.4. Let e, e 2 be edges of P. We say that e reaches e 2 withinh in k steps and in t ways if there exist at least t internally disjoint hyperpaths inh of length k from e to e 2. For t= we will skip the phrase in t ways. For an edge e P, we denote by

7 6 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS Four + (e,h) the set of those edges of P, which are reached from e withinh in four steps and inγ 0 n ways, and by Four (e,h) the set of all edges of P which reach e withinh in four steps and inγ 0 n ways (see Figure 2), where γ 0 = α4 5000l 7. V k V j V i g e h FIGURE 2. The fourth neigborhoods of e (g Four (e,h), h Four + (e,h)). Let us now provide some intuition for why it is necessary to consider the fourth hypergraph neighborhood of a graph edge. Suppressingα,δ,ɛ, most edges of P belong to about n/l 2 triplets ofh (see Fact 3.2), but any such edge e can be completely cut off from the rest ofh if no stronger assumption is made. Indeed, the total number of triplets extending triplets containing e is of the order n 2, and clearly the removal of such a tiny fraction of triplets cannot affect the δ-regularity which controls only sets of hyperedges of size, roughly, n 3 /l 3. In two steps, only about n 2 /l 4 edges are reached from a typical edge. Most of them extend to about n/l 2 triplets, a total of n 3 /l 6 still much less than n 3 /l 3 if l is large. To estimate the number of edges reached from a typical edge in three steps, the quantity n 3 /l 6 has to be divided, due to repetitions, by, roughly, n/l 4 (the number of vertices forming triangles with two given, disjoint edges), yielding only n 2 /l 2 edges. Again, they belong to about n 3 /l 4 δn 3 /l 3 triplets a quantity not under control. Hence, the shortest distance at which a typical edge can reach a substantial number of other edges is four. Theorem 3.5 below states that, indeed, most edges have large fourth neighborhood, and, more importantly, edges with large fourth neighborhood are mutually connected by short hyperpaths. Let us denote by R 0 (H)=R 0 the set of all edges of P, for which min ( Four + (e,h), Four (e,h) ) ( ) α 4 n 2 < 2000 l. Theorem 3.5. For all realα (0, ) andδ<δ 0, whereδ 0 is as in Theorem 2.9 there exist two sequences r(l) andɛ(l) such that for allh, P and for all integer l, if (H, P) is an (α,δ, l, r(l),ɛ(l))-triad with V = V 2 = V 3 =n sufficiently large, then (i) R 0 27 δn 2 /l, and

8 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 7 (ii) for every ordered pair of disjoint edges (e, f ) (P R 0 ) (P R 0 ) and for every set S V(H)\(e f ) of size S n/ log n, there is inh an S -avoiding hyperpath from e to f of length at most twelve. 4. TWOLEMMASANDMAINPROOFS Theorems 2.9 and 3.5 are straightforward consequences of two technical lemmas. A subgraph A of P=P 2 P 23 P 3 is called framed if for some i< j 3, A P i j. Our first lemma needs only the assumption that (H, P) is an (α,δ, l, r(l),ɛ(l))-triad, where r. Lemma 4.. For all c (0, ) andα (0, ) and for allδ<δ, where δ = αc , there exists a sequenceɛ(l) so that for allh, P and integer l if (H, P) is an (α,δ, l,,ɛ(l))- triad with V = V 2 = V 3 =n sufficiently large, then the following is true: For every subgraph P P, where P 29 δn 2 /l, and for every pair of framed subgraphs A and B of P P, each of size at least cn 2 /l, there exist edges a A and b B and a hyperpath in H P from a to b of length at most four. Our second lemma asserts that for a typical (H, P), apart from a small set of edges P 0, all other edges of P have their fourth neighborhood substantial, even if the edges of P 0 are to be avoided. This lemma needs the whole strength of the (δ, r)-regularity. Lemma 4.2. For all realα (0, ) and for allδ<δ 2, where δ 2 = α2 80 2, there exist two sequences r(l) andɛ(l) such that for allh, P and integers l if (H, P) is an (α,δ, l, r(l),ɛ(l))-triad with V = V 2 = V 3 =n sufficiently large, then there exists P 0 P, P 0 27 δn 2 /l, such that (4) min( Four + (e,h P 0 ), Four ) ( ) (e,h P 0 ) α 4 n l for all e P P 0. From Lemmas 4. and 4.2 we immediately derive our main result. Proof of Theorem 2.9. Note that for c=α 4 /3000,δ 0 =δ <δ 2. Givenαandδ<δ 0, letɛ (l) satisfy Lemma 4. with c= α4, and let sequences r(l), andɛ (l) satisfy Lemma 4.2. We claim that Theorem 2.9 is true with the above choice of r(l) and withɛ(l)=min(ɛ (l),ɛ 2 (l)). Indeed, consider anyh, P and l such that (H, P) is an (α,δ, l, r(l),ɛ(l))-triad and apply Lemma 4.2. It follows that there exists P 0 P, P 0 27 δn 2 /l, such that (4) holds for all e P P 0. Fix disjoint e, f P P 0, and a set S V(H)\(e f ) of size S n/ log n.

9 8 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS Define P S ={e P:S e } and observe that P S =o(n 2 ), and thus, for large n, P 0 P S 29 δn 2 /l, and min ( Four + (e,h P 0 ), Four (e,h P 0 ) ) ( ) α 4 n 2 P S 3000 l. P 2 e a P b P 3 f A B Four + (e,h P 0 ) Four ( f,h P 0 ) 4+4+4=2 FIGURE 3. A hyperpath from e to f. (An illustraction of the proof of Theorem 2.9) Since (H, P) is also an (α,δ, l,,ɛ (l))-triad, we may apply Lemma 4. with c= α to A=Four + (e,h P 0 )\ P S, B= Four ( f,h P 0 )\ P S and P = P 0 P S, obtaining edges a A and b B, and a hyperpathp inh (P 0 P S ) from a to b of length at most four. Let I= V(P ) f\ a. Among at leastγ 0 n> I S (for large n) internally disjoint hyperpaths from e to a inh P 0 choose one which is disjoint from I S, obtaining an S -avoiding hyperpathp 2 inh P 0 from e to b of length at most eight. Finally, set J= V(P 2 )\b and choose a hyperpathp 3 inh P 0 from b to f which avoids the vertices of J S. This way we obtain an S -avoiding hyperpath inh P 0 from e to f of length at most twelve (see Figure 3). Proof of Theorem 3.5. Since R 0 P 0, where P 0 is as in Lemma 4.2, part (i) follows from the estimate on P 0. The proof of part (ii), is very similar to that of Theorem 2.9. We define P S as before and apply Lemma 4. with c= α to A=Four + (e,h)\ P S, B= Four ( f,h)\ P S and P = P S, obtaining edges a A and b B, and a hyperpathp inh P S from a to b of length at most four. Finally, we extendp to an S -avoiding hyperpath inh.

10 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 9 Remark 4.3. It will follow from the proof of Lemma 4. that, in fact, depending on the position of the sets A and B, the promised hyperpath is precisely of length two, three or four. Consequently, depending on the position of e and f, the length of a hyperpath from e to f, guaranteed by Theorems 2.9 and 3.5, is precisely ten, eleven or twelve. 5. SHORTPATHSBETWEENLARGESETSOFEDGES In this section we prove Lemma 4.. We begin with formulating a claim from which the lemma will follows quite easily. Let E be any framed subgraph of P. Then First + (E,H) and S econd + (E,H) denote the sets of all edges h P reached inh by an edge g E in one and, respectively, in two steps. Sets First (E,H) and S econd (E,H) are defined similarly, by replacing the phrase reached inh by an edge g E by reaching inh an edge g E. Throughout, i jk always stands for any one of the sequences: 23 or 23 or 32, that is, sequences which follow the cyclic ordering 23. Claim 5.. For all c (0, ) andα (0, ), all 0<δ<min(α, c 6 /50 8 ) and sequences 0<ɛ(l) δ, and all integers l, if (H, P) is an (α,δ, l,,ɛ(l))-triad with V 0l 3 = V 2 = V 3 =n sufficiently large, then for all P P of size P 29 δn 2 /l and for all sets E P i j P of size E cn 2 /l, (5) min ( First + (E,H P ), First (E,H P ) ) c n 2 6 l, (6) min ( S econd + (E,H P ), S econd (E,H P ) ) ) ( 4δ/8 n 2 c l, In order to derive Lemma 4. from Claim 5. we need one more simple fact about vertexdisjoint subgraphs of bipartite graphs. Fact 5.2. Let A and B be two bipartite graphs with the same bipartition V V 2, V = V 2 =n. Then there exist A A and B B such that A A 2 2 2(A), B B and 2 V(A ) V(B ) V 2 =, where 2 (A) is the maximum degree in A among the vertices of V 2. Proof. Let us put vertices of the set V 2 in two lines: one ordered by their degrees in A in the descending manner (line A), the second the same with respect to B (line B). Now include the first vertex on line B to B and remove it from both lines. We repeat this step for line A and then again for B and so on until all vertices are placed in one of the sets A or B. (Note that V(A ) = n/2 and V(B ) = n/2.) Along the way, let us match each vertex of B included to B with the one included to A in the very next step (if n is odd, the vertex included to B last remains unmatched). Because we have started with the vertex of the largest degree in B, its match has a smaller or equal degree in B, and this is true for each matched pair. Therefore, we have B B. To prove 2 that A A 2 2 2(A) we apply the same reasoning to the set A minus all edges incident to the first vertex included to B. Proof of Lemma 4.. Given c andα, let (7) δ<δ = αc <α(c/3)6 50 8

11 0 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS and Note thatδ<α, and δ ɛ(l)= 0l 3 (8) 4δ 8 c/3 + c 2 > +ɛ(l)l, the latter by inequalitiesδ 8 < c c/(50 3) andɛ(l)l<c/50 2. LetH, P, l, A, B, and P be as in Lemma 4.. Without loss of generality we assume that A P 2 and will consider all three cases for B. If B P 3, apply Claim 5. with E=A to obtain a set A 3 = S econd + (A,H P ) P 3 P of at least 4δ 8 n2 c l edges. By (3) and (8), we conclude that B A 3, implying the existence of a hyperpath withinh P from an edge a A to an edge b B of length two. If B P 2, we use Fact 5.2 to obtain two subgraphs A A and B B such that A (c/3)n 2 /l (for n sufficiently large), B (c/2)n 2 /l and V(A ) V(B ) V 2 =. Then by Claim 5. applied with c c/3, the set A 3 = S econd + (A,H P ) P 3 P has cardinality at least 4δ 8 n2 c/3 l, and taking B 3 = First (B,H P ) P 3 P, by Claim 5. applied with c c/2, we have B 3 c n 2 2 l. Again, by (3) and (8), we conclude that B 3 A 3. Let zu B 3 A 3 and let xyzu and zuv be hyperpaths, respectively, from a= xy to uz and from zu to b=vu. Note that by the disjoint choice of A and B we have y v, and so xyzuv is a hyperpath withinh P from a A to b B of length three. The last case is when B P 23. Here also we apply Fact 5.2 to obtain two subgraphs A A and B B such that A (c/3)n 2 /l, B (c/2)n 2 /l and V(A ) V(B ) V 2 =. (Technically, we identify for a moment sets V and V 3 to treat A and B as two bipartite graphs on the same vertex set.) By Claim 5. applied with c c/3, the set A 3 = S econd + (A,H P ) P 3 P consists of at least 4δ 8 n2 c/3 l edges, and taking B 3 = S econd (B,H P ) P 3 P, by Claim 5. applied with c c/2, we get B 3 4δ 8 n2 c/2 l.

12 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI Again, by (3), (8) and (7), we conclude that B 3 A 3. Let zu B 3 A 3 and let xyzu and zuvw be hyperpaths, respectively, from a= xy to uz and from zu to b=wv. Note that by the disjoint choice of A and B we have y v, and so xyzuvw is a hyperpath within H P from a A to b B of length four (see Figure 4). V 3 V 2 B b w v V V 3 A 3 z u B 3 V 2 V a y x A FIGURE 4. An illustraction of the last case of the proof of Lemma 4.. It remains to prove Claim 5.. We first show a simple but crucial fact which will be applied twice in the proof of Claim 5.. Fact 5.3. For any realα,δ (0, ), integer l andɛ δ/(0l 3 ), let (H, P) be an (α,δ, l,,ɛ)-triad. Let, further, A V j, B V i and Q P be such that P ki Q 29 δn 2 /l, A an, B bn, and every vertex of A has in Q at leastβ B /l neighbors in B and at leastγn/l neighbors in V k. If min(γ, bβ)>ɛl and abβγ 43 δ then H Tr(Q) > 2(α δ)δn 3 /l 3. Proof. By the (/l,ɛ)-regularity of P ki, we have ( Tr(Q P ik ) deg Q (y, B)deg Q (y, V k ) ) l ɛ y A 90 abβγn3 l 3. On the other hand, setting Q = P i j P jk (P ki Q), by Corollary 8.4, we have Tr(Q ) <29 δ(.2) n3 l 3+ 4ɛn3 < 36 δ n3 l 3, and so, by our assumptions and Fact 8.6, we may estimate ( 9 Tr(Q) Tr(Q P ki ) Tr(Q ) 0 43 δ 36 ) n 3 δ l > 3 2δn3 l 3 Therefore, by the (α,δ, )-regularity ofh, H Tr(Q) d H (Q)= >α δ. T r(q) >δ Tr(P).

13 2 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS Proof of Claim 5.. By symmetry, it is enough to prove only that First + (E,H P ) (c/6)n 2 /l and similarly, that S econd + (E,H P ) ( 4δ /8 / c ) n 2 /l. Let us fixαand c, 0<α, c<, and let (9) δ<min ( α, ) c Further, withδgiven above, let for all l δ (0) ɛ(l) δ 4 0l 3< l c < c 20l. LetH, P, and l be as in Claim 5.. Setɛ=ɛ(l) for convenience and fix i< j 3. Let E be a set of at least cn 2 /l edges of P i j P. Define E ={yz P jk : xyz H and xy E and xz P for some x V i } and assign to each edge yz of E one (arbitrary) vertex x= x yz V i which together with yz satisfies the conditions in the definition of E. Finally, let E 2 = {zw P ki : w x yz and yzw H and yw P }. yz E P Note that E P = First + (E,H P ), and that by avoiding w = x zy, E 2 P S econd + (E,H P ). Observation 5.4. Trivially, if xy E, yz P jk E and xz P then xyz H. Similarly, but more subtly, if yz E P, zw P ki E 2, and yw P, then yzw H unless w= x yz, which implies that the edges of E P, P ki E 2 and P i j P span at most E P hyperedges inh. Using these observations and Fact 5.3 we will first show that a significant fraction of vertices y V j have large (close to n/l) neighborhood in E, and so subgraph E P is large. Then we will argue that most vertices of V k have large degree in E 2, meaning that the set E 2 must be very large (close to n 2 /l), and so must be E 2 P. Let { ( ) } L 0 = y V j : deg P jk(y)< l ɛ n. By (9) with A=V k, we have L 0 ɛn. Next, let us consider the set { L= y V j L 0 : deg E (y) cn }. 2l Observe that L cn/3. Indeed, otherwise, using (8) and (0), we obtain a contradiction ( ) ( ) E < L n l +ɛ + L 0 n l +ɛ +ɛn 2 + n cn 2l < cn2. l We proceed with the following fact. Set E = P jk E and L = y L:deg E (y)>7 δ 4 n c l.

14 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 3 Fact 5.5. L 3 δ 4 c n Proof. Assume L >3(δ 4 / c)n and apply Fact 5.3 with A=L, B=V i (and so b=), β=c/2, a=3δ 4 / c, andγ=7δ 4/ c to the 3-partite subgraph Q=Q i j Q jk Q ki, where Q i j = E[V i, L ], Q jk = E [L, V k ] Q ki = P ki P. As min(γ, bβ)>ɛl and abβγ>43 δ, it follows thath Tr(Q). However, by the construction of Q (see Observation 5.4) we haveh Tr(Q)=. This contradiction ends the proof of Fact 5.5. We set L = L L. By (9), () L = L L 3 cn 3δ 4 c n> 4 cn>ɛn. Note that every vertex y L has deg E (y)> n l ɛn 7δ 4 c n l, and thus, by (9), (0) and (), E L n l ɛn 7δ 4 n > c c l 5 To complete the proof of the inequality (5) we count the number of edges in E P the latter by (9). E P > c n 2 29 δ n2 5 l l n 2 l. > c n 2 6 l, We continue with the proof of the inequality (6). Let E 2 = P ki E 2. Note that by the (/l,ɛ)-regularity of P jk and P ki, Fact 8.3 and (), the set { ( ) ( ) } Ł 0 = z V k : deg P jk(z, L )< l ɛ L or deg P ki(z)< l ɛ n has size Ł 0 2ɛn. Next, let us consider the set Ł = z V k : deg E (z, L )>7 δ 8 l L. Since each vertex of L has in E degree at most 7(δ 4 / c) n, a simple, double counting l argument shows that Ł (δ 8 / c)n. Further, let Ł 2 = z V k : deg P (z, L )>6 δ 4 L cl.

15 4 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS Clearly, Ł 2 <(δ 4 / c)n, since otherwise P >29 δn 2 /l a contradiction. Set Ł= V k \ (Ł 0 Ł Ł 2 ) and define Ł = z Ł:deg E 2 (z)>9 δ 4 n c l. Observe, by (9) and (0), that for all z Ł(and thus for all z Ł ) we have deg E P (z, L )> l ɛ 8 7δ 6 δ 4 L > 4 l cl 5l L. Fact 5.6. Ł 24 δ 4 c n. Proof. The proof of this fact is very similar to the proof of Fact 5.5. We will argue that the inequality Ł >24(δ 4 / c)n contradicts a conclusion of Fact 5.3. Define a 3-partite subgraph Q=Q i j Q jk Q ki as follows: Q i j = P i j P, Q jk = E [Ł, L ] P, Q ki = E 2 [Ł, V i ]. By the construction of Q and Observation 5.4 we have H Tr(Q) E =O(n 2 ). Apply Fact 5.3 with A=Ł, B= L (and so b=c/4),β=4/5, a=24δ 4 / c, andγ=9δ 4/ c to yield H Tr(Q) =Ω(n 3 ). For large enough n, this is a contradiction which ends the proof of Fact 5.6. To complete the proof of the inequality (6), set Ł = Ł\Ł = V k \ (Ł 0 Ł Ł 2 Ł ) and note that for every vertex z Ł, by (0), we have deg E2 (z) n l ɛn 9δ 4 n c l > ( 0δ 4 ) n c l. Note also that all the exceptional sets Ł, Ł 0, Ł and Ł 2 contain together less than 2(δ 8 / c)n vertices and therefore Ł >( 2δ 8 / c)n. Thus, by (9), E 2 Ł 0 δ 4 n 3 c l > δ 8 n2 c l, hence E 2 P > 3 δ 8 n2 29 δ n2 c l l > 4 δ 8 c n2 l.

16 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 5 6. THE FOURTH NEIGHBORHOOD In this section we prove Lemma 4.2. Let us begin with some heuristic. We call an edge H-good, or just good if, say, Γ H (e) 2 9 αn/l2. As proved in [6] (see Fact 3.2 above), for most edges of P we have Γ H (e) αn/l 2, so most edges are good, but unfortunately, some of these good edges may have small fourth, and even second neighborhood. Indeed, it might happen that for a good edge e= xy, whenever xyz Hthen yz has a very small neighborhood. To find a large subset of good edges e with large fourth neighborhoods Four + (e,h) and Four (e,h), one could argue as follows. Suppose that the set of bad (=not good) edges has sizeρn 2. Then, for each i=, 2, 3, at most ρn vertices of V i are incident to at least ρn bad edges (let us call these vertices bad), and, provided ρ /l 2, one could start at a good edge with good endpoints and move four steps, avoiding both, bad edges and bad vertices. The problem is that Fact 3.2 yields onlyρof order /l too large for our needs. To get around this problem we will find a subhypergraphh Hwith much less bad edges. This sounds paradoxical, since removing hyperedges can only decrease Γ H (e). Note, however, that edges e withγ H (e)= are not so bad there is no way to get to them! Let us call them H-dead. To distinguish between H-dead and other bad edges, we will alter our previous definition and call an edge e PH-bad if 0< Γ H (e) < 2 9 α n l 2. So, for anyh H, every edge of P is eitherh -good orh -bad orh -dead. Let us denote these three subgraphs by G H, B H and D H. For technical reasons we distinguish also a class F 0 of atypical edges of P. For all i< j 3, an edge e P i j belongs to the subgraph F 0, if either it is not typical or at least one of its ends is not typical in P i j. Note, that by Fact 8.3 and Corollary 8.4, F 0 <24ɛn 2. Claim 6.. For allα (0, ) and for allδ<α/9 2 there exist two sequences r(l) andɛ(l) so that for allh, P and integer l if (H, P) is an (α,δ, l, r(l),ɛ(l))-triad then there exists a subhypergraphh H such that B H δn 2 /l 4, D H <22 δn 2 /l and F 0 D H. Proof of Lemma 4.2. With givenαand δ<δ 2 = α2 80 2< α 9 2, let r(l) andɛ (l) be such that Claim 6. holds. Set ( ɛ(l)=min ɛ (l), α 4 20, 000l 8 We will prove Lemma 4.2 with this choice of sequences r(l) andɛ(l). Given integer l, let a pair (H, P) be an (α,δ, l, r,ɛ)-triad, where r=r(l) andɛ=ɛ(l). Further, leth be as in Claim 6., let V = {v V : deg BH (v) δ n }, l 2 G H ={e G H : e V }, and let P 0 = B H D H G H. ).

17 6 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS Note that F 0 P 0. It remains to prove two facts about P 0. Fact 6.2. P 0 27 δn 2 /l Proof. To prove this fact, note that V 2 δn/l 2, and so G H 2n V 4 δn 2 /l 2. Therefore P 0 B H + D H + G H δ n2 l δ n2 + 4 δ n2 l l 2 27 δ n2 l. Fact 6.3. For every edge e P P 0 the inequality (4) holds. Proof. By symmetry, we will only prove that Four + (e,h P 0 ) ( α 4 /2000 ) n 2 /l. Without loss of generality we may assume that e= xy P 2 P 0, where x V. Then, by our choice ofδ, the set of vertices z, such that xyz H and yz, xz P 0, has size at least 2 (2) 9 α n l 2 4 δ n n l 2>α 5 l 2, where the deletion takes care of all z V, as well as all z with yz or xz in B H (clearly, yz and xz cannot beh-dead). Thus, xyz H P 0. For each such z, the edge yz belongs in turn to at leastαn/(5l 2 ) triplets yzw H with w V \{x}, yw P 2 P 0 and zw P 3 P 0. So, altogether there are at leastα 2 n 2 /(25l 4 ) edges of P 3 P 0 reached (withinh P 0 ) in two steps by e. Repeating this argument again we obtain at leastα 4 n 4 /(625l 8 ) hyperpaths xyzwuv H P 0 of length four originating at e= xy. Let us estimate, by counting repetitions, how many different edges uv P 23, u V 2, v V 3, are indeed reached by e in four steps (and in many ways). Consider an auxiliary bipartite graph C= (X, Y, E C ), where X=E(P 3 ), Y= E(P 23 ), and{zw X, uv Y} E c if xyzwuv is a hyperpath inh P 0. Hence, E C α 4 n 4 /(625l 8 ). Every hyperpath xyzwuv must satisfy that (see Figure 5). V 2 V 3 V z N P 23(u, N P (xy)) and w N P (uv, N P 2(y)) N P 23(u, N P (xy)) v z P 23 y u P 3 P 2 e w x N P (uv, N P 2(y)) FIGURE 5. A hyperpath of length four originating at e= xy. Since F 0 P 0, therefore xy F 0, so we haveɛn< N P 2(y) <(/l+ɛ)n andɛn< N P (xy) <(/l+ɛ) 2 n. By Fact 8.3, all but at mostɛn 2 edges uv satisfy N P 23(u, N P (xy)) <

18 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 7 (/l+ɛ) 3 n and all but at most 3ɛn 2 edges uv satisfy N P (uv, N P 2(y)) <(/l+ɛ) 3 n. If both these sets are greater thanɛn then, by the (/l,ɛ)-regularity of P 3, there are at most (/l+ɛ) 7 n 2 edges zw P 3 between them. Otherwise, the number of such edges zw is at mostɛn 2 < n 2 /l 7, the last inequality by our assumption onɛ. Thus, (/l+ɛ) 7 n 2 < 20 9 n2 /l 7 is an upper bound on the degree in C of all but at most 4ɛn 2 edges uv whose degree can be even equal to n 2. Denote the set of such edges by Y and set = max e Y\Y deg C (e). Then 0 = 20 n 2 9 l 7. Let { Y 2 = uv Y : deg C (uv) } E C, 2 P 23 and Y 3 = Y\ (Y Y 2 ). We have E C Y n 2 E C + Y Y 3 2 P 23 4ɛn4 + Y E C 2. Therefore, by our choice ofɛ, ( ) α 4 n 4 Y 2 4ɛn4 250l8 > α4 n l Another words, at leastα 4 n 2 /2000l edges uv P 23 can be reached from e by no less than E C 2 P 23 > α4 n l 7 hyperpaths of length four, or equivalently, via that many edges zw P 3. It is easy to see that among these edges there is matching of size at least Proof of Claim 6.. α 4 n 5000l 7. Givenα, let δ< α 9 2, and letɛ (l) be such that Fact 3.2 holds with aboveαandδ. Further, let for all l, r(l)=32l 3 and ( δ ) (3) ɛ(l) = min ɛ (l),. 24l 3 We will prove Claim 6. with this choice ofδ, r(l) andɛ(l). Given integer l, let a pair (H, P) be an (α, δ, l, r, ɛ)-triad, where r = r(l) and ɛ = ɛ(l).

19 8 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS We will define a process of deleting hyperedges which after finitely many rounds will arrive at a subhypergraphh ofh satisfying the conclusions of Claim 6.. Recall that for an arbitrary hypergraphh and a graph G, we denote byh G the subhypergraph ofh obtained by removing all hyperedges containing at least one edge of G. The initial step of the procedure isolates all edges of F 0. SetH =H F 0. Clearly, for each e F 0, we haveγ H (e)= and so e ish -dead. In each next round we similarly kill edges of P which are bad in the current subhypergraph. For technical reasons these rounds take cyclically care of the edges of P 2, P 23, and P 3. For each s=, 4, 7,..., let F s ={e P 2 : e ish s -bad}, H s+ =H s F s, F s+ ={e P 23 : e ish s+ -bad}, H s+2 =H s+ F s+, F s+2 ={e P 3 : e ish s+2 -bad}, H s+3 =H s+2 F s+2. In each operation of the typeh s+ =H s F s we remove all hyperedges which contain H s -bad edges e of P 2, P 23 or P 3. Thus, those edges becomeh s+ -dead and therefore will never become bad again. It follows that all sets F s are disjoint, and, in paticular, for s, F s F 0 =. Our immediate goal is to estimate r s= F s. Let us define 3-partite subgraphs Q s of P, s=, 2,..., r, as follows: If F s P i j then s s Q s = F s Pik F t P jk F t, t= SetQ=(Q,..., Q r ). Observe that for slightly enlarged subgraphs Q s = F s P ik P jk (where F s P i j ), we have, by (3) and the fact that F s F 0 =, ( ) 2 Tr(Q s ) F s n l ɛ 3 n 4 l 2 F s. Trivially, r Tr(Q s ) s= r Tr(Q s ), but the reverse inclusion is also true. Indeed, for xyz Tr(Q s ) set t 0 = min{t :{xy, xz, yz} F t }. Then t 0 s and xyz Tr(Q t0 ). Moreover, because the sets F s are disjoint (and so are Tr(Q s )), r r r Tr(Q s ) = Tr(Q s ) = Tr(Q s ) r Tr(Q 3 s ). Hence, s= (4) s= s= s= r Tr(Q s ) n 4 l 2 s= r F s. s= t= s=

20 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 9 On the other hand, however, by the definition of anh s -bad edge, for all s r, forcing H Tr(Q s ) < F s 2 9 α n l 2, d H (Q)< 8 9 α, where d H (Q) is defined in (). Therefore, by the (α,δ, r)-regularity ofh, r (5) Tr(Q s ) δ Tr(P), s= since otherwise d H (Q)>α δ 8 α. This inequality together with (3), (4) and Fact implies that (6) r F s 4 s= r s= Tr(Q s ) l2 n < 8δn2 l. Thus, more than a half of the sets F s, s r, have size F s 6δn 2 /lr, and so two consecutive sets must be such, that is, there exists an index s r 2, such that max( F s+, F s+2 ) 6δ n2 lr = 2 δn2 l 4. Let s 0 be the smallest index s with this property. Without loss of generality we may assume that F s0 P 2. SetH =H s0 +. Observe that there is noh -bad edge in the graph P 2, while in each P 23 and P 3 we have at most 2 δn2 /l 4 H -bad edges. In fact, the set ofh -bad edges is the union of F s0 + (H -bad edges in P 23 ) and of a subgraph of F s0 +2 (F s0 +2 may containh s0 +2-bad edges which were noth s0 +-bad). As for theh -dead edges, these are exactly the edges in s 0 i=0 F i plus all the edges e P which were originally dead, that is, which hadγ H (e)=. We have already estimated s 0 i= F i in (6), while, by (3), F 0 <24ɛn 2 <δn 2 /l. Finally, by Fact 3.2, there are no more than 2 δn 2 /l originally dead edges. Therefore we have D H < s 0 i= Hence, Claim 6. is proved. F i + F 0 + D H <8δ n2 l +δ n2 l 7. APPLICATIONS + 2 δ n2 l In this section we give two immediate applications of Theorem 2.9. < 22 δ n2 l 7.. Long hyperpaths. The Blow-up Lemma of Komlós, Sárközy and Szemerédi [4] states that with a suitable choice of parameters every s-partite graph G with s-partition V V s in which all bipartite subgraphs G[V i, V j ] are (d,ɛ)-regular contains all bounded degree s-partite graphs G with s-partition V V s, where for all i=,..., s, V i V i, V i <( f (ɛ)) V i.

21 20 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS So far no analogous results exist for 3-uniform hypergraphs. As a first step toward a hypergraph Blow-up Lemma, we derive from Corollary 3.5 a simple consequence which establishes the existence of an almost hamiltonian hyperpath in a quasi-random 3-graph. Proposition 7.. For all realα (0, ) and for allδ<(δ 0 /4) 4, whereδ 0 is as in Theorem 3.5, there exist two sequences r(l) andɛ(l) such that for allh, P and integers l if a pair (H, P) is an (α,δ, l, r(l),ɛ(l))-triad with V = n sufficiently large, then there is inh a hyperpath of length at least ( δ 4 )n. Proof. Givenα, letδ < (δ 0 /4) 4 and r = r(l) iɛ (l) be ensured by Theorem 3.5. ɛ=ɛ(l)=δ 4 ɛ (l). Observe, that Set (7) 27 4δ 4< 27 δ 0 < α Let a pair (H, P) be an (α,δ, l, r,ɛ)-triad. Suppose, that no hyperpath inh is longer than ( δ 4 )n. For a hyperpath Q, leth Q be the subhypergraph ofh, obtained by deleting from H all, but the last four vertices of the path Q (if V(Q) <4, then we seth Q =H). Let us fix an arbitrary edge e={x, y} P R 0 and let Q be the longest hyperpath inh originating at e (in the cyclic order V V 2 V 3 V ) and such that its other endpair f P R 0 (H Q ). It follows trivially from the definition of the set R 0(H) that Q has at least four vertices. Let us denote the last four vertices of Q by x 3, x 2, x, x 0. Since V(Q) <( δ 4 )n, the subhypergraphh Q =H V(Q) has at leastδ 4 n vertices. Moreover, since Q traverses the sets V, V 2, V 3 in the cyclic order, the sizes of the sets V(H Q ) V i, i=, 2, 3, differ from each other by at most one. Hence (see [8], Fact 4.2), the pair (H Q, P[V(H Q )]) is an (α, 4δ 4, l, r,ɛ/δ 4 )-triad. Note that V(H Q ) = V(H Q ) +4, 4δ 4 <δ0 andɛ/δ 4 =ɛ (l). Therefore, by Theorem 3.5, V(H R 0 (H Q) 27 4δ 4 Q ) /3 2 l 27 4δ 4 V(H Q ) /3 2. l On the other hand, by the definition of R 0 (H Q ), we know that the edge f={x, x 0 } reaches in four steps at least α 4 V(H Q ) / l V(H > 27 4δ Q ) / n R 0 (H l Q) +2n other edges of P[V(H Q )] (the term 2n takes care of all edges with at least one endpoint in x 2 or x 3 ; the first inequality follows from (7) for large n). Therefore, there is at least one edge f ={x 3 x 4 } P[V(H Q )] R 0(H Q ), reached by f inh Q by at least three (in fact, many more) internally disjoint hyperpaths of length four of the form x x 0 x x 2 x 3 x 4. Thus, for at least one of them{x, x 2, x 3, x 4 } {x 3, x 2 }=, and we may extend Q by adding the vertices x, x 2, x 3, x 4 a contradiction with the maximality of Q (see Figure 6).

22 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 2 Q x 2 x 0 x 2 x 4 e f f x 3 x x H Q H Q x 3 FIGURE 6. A hyperpath Q originating at e. Similarly, one can prove that for most pairs of edges of P there is a path of length at least ( δ 4 )n between them Approximate decomposition into small diameter subhypergraphs. It is easy to see that for every n-vertex graph and for everyɛ> 0 one can partition E(G)=E 0 E k, where k /ɛ, so that E 0 ɛn 2 and for each i k the diameter of the subgraph G i = G[E i ] is at most 3/ɛ (see [7]). Thus, in a sense, every graph can be decomposed into a bounded number of small worlds provided a small set of edges can be ignored. Here both bounds, on the diameter and on the number of subgraphs G i depend linearly on /ɛ. Using the Szemerédi Regularity Lemma [9] and Corollary 8.5(b), one may put the cap of four on the diameter, at the cost of letting k, the number of subgraphs in the partition, to be an enormous constant. Proposition 7.2. [7] For allɛ> 0 there exist integers K and N such that for all n-vertex graphs G, where n N, there is a partition E(G)=E 0 E E k, where k K, and E 0 ɛn 2, and for each i k, the diameter of the subgraph G i = G[E i ] is at most four. An analogous result for 3-uniform hypergraphs follows from our Theorem 2.9 and the Hypergraph Regularity Lemma in []. Theorem 7.3. For allξ>0 there exist integers K and N such that for all n-vertex 3-uniform hypergraphsh, where n N, there is a partitionh=h 0 H k where k Kand H 0 ξn 3 and for each i k, every two pairs of vertices with positive degree inh i are connected by a hyperpath inh i of length at most twelve. Sketch of proof: Givenξ>0, set t 0 = 8/ξ,α=ξ/8 and letδ 0 > 0 and sequences r(l),ɛ 2 (l) be as in Theorem 2.9 with aboveα. Further, let N be the smallest natural number, for which Theorem 2.9 holds. Set ( ( ξ ) 2 ) δ=min δ 0, 6 and apply the Hypergraph Regularity Lemma of [] withδ,ɛ =δ 4,ɛ 2 (l) and r(l) to get T 0, L 0 i N 0. Set ( ( T0 K= )L 30 3 and N= max N 0, 6T ) 0, N T 0 ξ and leth be an arbitrary 3-uniform hypergraph with n Nvertices and H ξn 3 triplets.

23 22 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS By the Hypergraph Regularity Lemma, H admits an auxilary vertex partition V(H) = V 0 V V t, where t 0 t<t 0, V 0 <tand V = V 2 = = V t, and for each pair i, j, i< j t, partitions of the complete bipartite graphs K(V i, V j )= l a=0 P i a j, where l<l 0, such that most triplets ofh belong to (α,δ, l, r(l),ɛ 2 (l))-triads (H, P), where H =H Tr(P) and P=(P hi a, P h j b, Pi c j ), i< j<h t, a, b, c l. Let (H s, P s ), s=,...,k ( t 3) l 3 < K, be all such triads. For each s=,...,k, let (P s ) 0 be the subgraph of P s guaranteed by Theorem 2.9, and seth s =H s (P s ) 0. Then, each pair of edges of P s (P s ) 0, that is each pair of edges of P s with positive degree inh s is connected inh s by a hyperpath of length at most twelve. Let us seth 0 =H\ k s=h s. To complete the proof of Theorem 7.3, it remains to show that H 0 ξn 3. We omit the details of tedious but straightforward calculations. 8. APPENDIX ɛ-regular PAIRS Let G = (V, E) be a graph, where V and E are the vertex-set and the edge-set of G. Throughout the paper we often identify G with its set of edges and therefore write G instead of E. When U, W are subsets of V, we define For nonempty and disjoint U and W, e G (U, W)={{x, y} E:x U, y W}. d G (U, W)= e G(U, W) U W is the density of the graph G between U and W, or simply, the density of the pair (U, W). Definition 8.. Givenɛ> 0, a bipartite graph G with bipartition (V, V 2 ), where V =n and V 2 =m, is calledɛ-regular if for every pair of subsets U V and W V 2, U > ɛn, W > ɛm, the inequalities d ɛ< d G (U, W)<d+ɛ hold for some real number d>0. We may then also say that G, or the pair (V, V 2 ), is (d, ɛ)-regular. Let a graph G=(V, E) be given. We write N G (v) for the set of neighbors of v Vin the graph G. The size of N G (v) is N G (v) =deg G (v), the degree of v. We set N G (xy)= N G (x) N G (y) as the set of common neighbors of x, y V in G. For a set U V, we write N G (v, U) for the set of neighbors of v in U and N G (xy, U) for the set of common neighbors of x and y in U. The size of N G (v, U) is N G (v, U) =deg G (v, U). Definition 8.2. Let G=(V V 2, E) be a (d,ɛ)-regular bipartite graph, where V = V 2 = n. We say, that a vertex x V i, i=, 2, is typical in G, if the following inequalities hold n(d ɛ)<deg G (x)<(d+ɛ)n. Further, let G= G 2 G 23 G 3 be a 3-partite graph with partition (V, V 2, V 3 ), where V = V 2 = V 3 =n, and each graph G i j is (d,ɛ)-regular, i< j 3. We call a pair of vertices (x, y) V i V j typical if it satisfies inequalities n(d ɛ) 2 < N G (xy) <n(d+ɛ) 2.

24 JOANNA POLCYN, VOJTECH RÖDL, ANDRZEJ RUCIŃSKI AND ENDRE SZEMERÉDI 23 The next fact is well-known and follows immediately from Definition 8. (see e.g. [6]). Fact 8.3. For all realɛ> 0 and d>0, and for all integers n and m, the following holds. Let G be a (d,ɛ)-regular bipartite graph with a bipartition (V, V 2 ), where V =n, V 2 =m. Further, let A V 2, A >ɛm. Then all but at mostɛn vertices x V satisfy (8) deg G (x, A)<(d+ɛ) A, and all but at mostɛn vertices x V satisfy (9) deg G (x, A)>(d ɛ) A. In particular, if V = V 2 =n, then for each i {, 2}, all but at most 2ɛn vertices x V i are typical in G. Corollary 8.4. For allɛ> 0 and d>2ɛ and for all integers n, the following holds. Let G= G 2 G 23 G 3 be a 3-partite graph with partition (V, V 2, V 3 ), where V = V 2 = V 3 =n and each graph G i j is (d,ɛ)-regular, i< j 3. Then all but at most 4ɛn 2 pairs of vertices (x, y) V i V j are typical. Another simple consequence of Fact 8.3 deals with the distances in a quasi-random bipartite graph (see [6] and [7]). Corollary 8.5. Let B be a (d,ɛ)-regular bipartite graph with bipartition (V, V 2 ), where V = V 2 =n. (a) If d>2ɛ then all pairs of vertices of B of degree at leastɛn can be connected by paths of length at most four. (b) If d>4ɛ then by removing from B at most 2ɛn vertices (those of degree less than 3ɛn<(d ɛ)n), we obtain a subgraph with diameter four. Finally, we state another well-known result which tightly estimates the size of T r(g), the set of triangles in a quasi-random 3-partite graph G (see, e.g., [8]). Fact 8.6. Let G= G 2 G 23 G 3 be a 3-partite graph, where all three bipartite graphs G i j are (d,ɛ)-regular, i< j 3. If d>2ɛ then (d 3 0ɛ)< Tr(G) V V 2 V 3 < (d3 + 0ɛ). In particular, ifɛ< 0.d 3 then Tr(G) <2d 3 V V 2 V 3. REFERENCES [] Frankl, P., Rödl V., Extremal problems on set systemns, Random Struct. Algorithms 20 (2002) [2] Haxell, P. et al., The Ramsey number for hypergraph cycles II, in preparation. [3] Haxell, P. et al., The Ramsey number for hypergraph cycles I, submitted [4] Komlós, J., Sárközy, G., and Szemerédi, E., The blow-up lemma, Combinatorica 7 (997) [5] D. Kuhn and D. Osthus, in preparation. [6] Polcyn, J., Short paths in 3-uniform quasi-random hypergraphs, Discus. Math. Graph Theory 24 (2004) to appear. [7] Polcyn, J. et al., Holes and short paths in quasi-random pairs, in preparation. [8] Rödl, V., Ruciński, A., and Szemerédi, E., A Dirac-type theorem for 3-uniform hypergraphs, submitted.

25 24 SHORT PATHS IN QUASI-RANDOM TRIPLE SYSTEMS WITH SPARSE UNDERLYING GRAPHS [9] E. Szemerédi, Regular partitions of graphs. Problemes combinatoires et theorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 976), pp , Colloq. Internat. CNRS, 260, CNRS, Paris, 978. A. MICKIEWICZ UNIVERSITY POZNAŃ, POLAND EMORY UNIVERSITY, ATLANTA, GA A. MICKIEWICZ UNIVERSITY POZNAŃ, POLAND RUTGERS UNIVERSITY, NEW BRUNSWICK