Tight Hamilton cycles in cherry quasirandom $3$-uniform hypergraphs

We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so called {\em two-path} or {\em cherry}-quasirandom $3$-graphs. Our first result asserts that for any fixed real $\alpha>0$, cherry-quasirandom $3$-graphs of sufficiently large order $n$ having minimum $2$-degree at least $\alpha (n-2)$ have a tight Hamilton cycle. Our second result concerns the minimum $1$-degree sufficient for such $3$-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every $d,\alpha>0$ satisfying $d + \alpha>1$, any sufficiently large $n$-vertex such $3$-graph $H$ of density $d$ and minimum $1$-degree at least $\alpha \binom{n-1}{2}$, has a tight Hamilton cycle.

§ 1 . I n t ro d u c t i o n A theorem of Dirac [11] asserts that an n-vertex (n ≥ 3) graph whose minimum degree is at least n/2 contains a Hamilton cycle; moreover, the degree condition imposed here is best possible. A rich and extensive body of work now exists concerning the extent to which Dirac's result can be extended to uniform hypergraphs see, e.g., [4,8,9,13,14,15,16,17,19,20,22,23,33,34,35,36, 38] 1 . Allow us to not reproduce here the intricate development of these results as outstanding accounts of these already exist in the excellent surveys [24, 32,44].
We confine ourselves to 3-uniform hypergraphs (3-graphs, hereafter). A 3-graph C is said to form a loose cycle if its vertices can be cyclically ordered such that each edge of C captures 3 vertices appearing consecutively in the ordering, every vertex is contained in an edge, and any two consecutive 2 edges meet in precisely one vertex. We say that C forms a tight cycle if there exists a cyclic ordering of its vertices such that every 3 consecutive vertices in this ordering define an edge of C; this particularly implies that any two consecutive edges meet in precisely 2 vertices. deg H (u, v).
Resolving a conjecture of [19], first approximately [35] and then accurately [38], the latter result asserts that a sufficiently large n-vertex 3-graph H satisfying δ 2 (H) ≥ n/2 contains a tight Hamilton cycle. A construction appearing in [19] demonstrates that the codegree condition imposed here is best possible. Finding the correct threshold for δ(H) at which a 3-graph H admits a tight Hamilton cycle remained elusive for quite some time though. The problem has come to be known as the 5/9-conjecture [32, Conjecture 2.18] asserting that sufficiently large n-vertex 3-graphs H satisfying δ(H) ≥ (5/9+o(1)) n−1 2 admit a tight Hamilton cycle. Constructions appearing in [32,33] establish that the degree condition appearing in this conjecture is (asymptotically) best possible. The authors of [7] established that such 3-graphs admit a tight cycle covering all but o(n) of the vertices. Then, in a major breakthrough [30] (preceded by the deep result of [34] and around the same time as [7]), the 5/9-conjecture has been resolved.
An additional result relevant to our account is that of [27]. Presentation of the latter requires a brief overview regarding quasirandom 3-graphs. Launched in [5,41,42], the study of quasirandom graphs has developed into a rich and vast theory, see, e.g. [21]. While a canonical definition of quasirandom graphs was already captured in [5,41,42], for hypergraphs the pursuit after a definition extending [5] took much longer. An elaborate account regarding the development of this pursuit can be seen in [1,6,25,26,43] and references therein. Only recently with the work of [43] has this pursuit came to an end; an alternative combinatorial approach to the functional analytic work of [43] appears in [1].
Roughly speaking, for k ≥ 3 each set system of [k] = {1, . . . , k} forming a maximal anti-chain gives rise to a notion of quasirandomness for k-graphs. In the case of interest to us, that is k = 3, each of the maximal anti-chains 1 The study of perfect matchings in hypergraphs is intimately related to the Hamiltonicity problem. We omit references to such results as our work here was not directly influenced by this line of research. 2 Order of the edges inherited from the ordering of the vertices. defines a notion of quasirandomness referred to as * -quasirandomness with * ∈ { , , , }, respectively (concrete definitions follow below); here these notions are arranged from left to right in increasing order of strength so to speak.
A solid understanding of -quasirandomness (i.e., the weakest notion) was attained in [6,25]. More generally, we now know from [1,43] (and owing much to [26]) that all these notions are well-separated and form a certain hierarchy with -quasirandomness at the "bottom" as the weakest notion (so it forms the broadest class of hypergraphs). In what follows, however, we will not be bothered with these notions of quasirandomness per se. Instead we shall consider weaker related notions. Borrowing notation from [28,29], given d, ∈ (0, 1], an n-vertex 3-graph H is said to be ( , d) -dense if e H (X, Y, Z) := |{(x, y, z) ∈ X × Y × Z : {x, y, z} ∈ E(H)}| ≥ d|X||Y ||Z| − n 3 (1.1) holds for every X, Y, Z ⊆ V (H). If and d exist yet are not made explicit, then we say that H is -dense. The notion of -quasirandomness comes about if one imposes on e H (X, Y, Z) the upper bound corresponding to (1.1).
Returning to Hamiltonicity, one encounters the following remarkable result of [27] stated here for 3-graphs only.
Theorem 1.2. [27] For every d, α ∈ (0, 1] there exist an n 0 and a > 0 such that the following holds whenever n ≥ n 0 and even. Let H be an n-vertex ( , d) -dense 3-graph satisfying δ(H) ≥ α n−1 2 . Then, H admits a loose Hamilton cycle. Theorem 1.2 settles the issue of emergence of loose Hamilton cycles in quasirandom 3-graphs for any notion of quasirandomness and any type of degree (the latter owing to [32,Remark 1.4]). It asserts that all minimum degree conditions sufficient for the emergence of loose Hamilton cycles in quasirandom 3-graphs are degenerate (i.e., any positive α suffices) 3 .
For tight cycles, however, a result analogous to Theorem 1.2 does not exist for -quasirandom 3-graphs. Indeed, [27,Proposition 4] asserts that for every > 0 and a sufficiently large n, an n-vertex ( , 1/8) -quasirandom 3-graph H exists satisfying δ(H) ≥ (1/8 − ) n−1 2 and having no tight Hamilton cycle. The constant 1/8 here is not best possible though as the following construction demonstrates. Let n ∈ N be sufficiently large and let V = X∪Y be a set of n vertices such that |X| = 2n/3 + 1 and |Y | = n/3 − 1 (assume 3 | n). Let G ∼ G(n, p) be the random graph put on V where each edge is put in G independently at random with probability p; we determine p below. Define H to be the 3-graph whose set of vertices is V and whose set of edges consists of: • all the sets e ∈ V 3 satisfying G[e] ∼ = K 3 and e ⊆ X or e ⊆ Y or |e ∩ X| = 1; • together with the sets e ∈ V 3 satisfying 2 = |e ∩ X| := |{u, v}| and uv / ∈ E(G).
An argument similar to the one used in [33, Construction 2] asserts that H has no tight Hamilton cycle. Indeed, no tight path can connect a triple contained in X with a vertex of Y . Consequently, if H were to admit a tight Hamilton cycle C then X must be an independent set in C and Y a vertex-cover of C. This together with the fact that C is 3-regular (with respect to 1-degree, that is) we reach n = e(C) ≤ y∈Y deg C (y) = 3|Y | < n; a contradiction. Every triple e is taken into H either with probability p 3 or 1 − p. Insisting on p 3 = 1 − p, so that p = 0.6823. Using binomial tail estimations it follows that it is highly likely that H would have edge density ≈ 0.3177, satisfy δ(H) ≈ 0.245n 2 , and be -dense. We acknowledge the discussions [31] regarding this construction.
Replacing the degree condition seen in Theorem 1.2 with a codegree condition would be insufficient in order to yield a result analogous to Theorem 1.2. Indeed, in [27] it is indicated that an adaption of the construction seen in [27,Proposition 4] yields a -dense graph H with δ 2 (H) ≥ n/9 admitting no tight Hamilton cycle.
1.1 O u r re s u l t s If we were to "climb" up the hierarchy of notions of quasirandomness for 3-graphs and strengthen the quasirandomness condition satisfied by the host 3-graph would we then encounter an analogue of Theorem 1.2 for tight Hamilton cycles? Let d, If and d exist yet are not made explicit, then we say that H is -dense (pronounced cherry-dense).
Our first main result asserts that the minimum codegree condition sufficient to imply the emergence of a tight Hamilton cycle in -dense 3-graphs is degenerate. Theorem 1.4. For every d, α ∈ (0, 1], there exist an integer n 0 and a real > 0 such that the following holds for all n ≥ n 0 . Let H be an n-vertex ( , d) -dense 3-graph satisfying δ 2 (H) ≥ α(n−2). Then, H has a tight Hamilton cycle.
In Theorem 1.4, the parameter d plays a somewhat docile role in the sense that no strict conditions other than it being fixed and positive need be imposed. For our second result, we consider -dense 3-graphs with an imposed minimum 1-degree condition. Here, a condition on d arises (for us) as follows.
Theorem 1.5. For every d, α ∈ (0, 1] satisfying α + d > 1, there exist an integer n 0 and a real > 0 such that the following holds for all n ≥ n 0 . Let H be an n-vertex ( , d) -dense 3-graph satisfying δ(H) ≥ α n−1 2 . Then, H has a tight Hamilton cycle. Unlike Theorem 1.4, the requirement α + d > 1 does not allow for a degenerate minimum 1-degree condition. Nevertheless, it is more flexible than other results mentioned thus far. We conjecture (with some hesitation) that the condition α + d > 1 appearing in Theorem 1.4 can be replaced with degenerate conditions for both α and d as follows.
"Between" -quasirandomness and -quasirandomness, there lies -quasirandomness. For d, holds for every P ⊆ V (H) × V (H) and every X ⊆ V (H). Unlike -quasirandom 3-graphs, for which the Turán density of K 4 (the complete 3-graph on 4 vertices) is zero [28], the Turán density of K 4 with a single edge removed) in -quasirandom 3-graphs is 1/4 [29]. The absorbing configurations (see § 4 for details) used in this account involve copies of K . Consequently results in the spirit of Theorems 1.4 and 1.5 cannot possibly be attained for -quasirandom 3-graphs using the absorbing-path method and the absorbing configurations used in our account. We subscribe to the point of view that the flaw is not in the method and that for -quasirandom 3-graphs the minimum 1-degree and 2-degree conditions sufficient to imply tight Hamiltonicity are both non-degenerate. The fact that the Turán density of K (3)− 4 in -quasirandom 3-graphs coincides with that seen in -quasirandom 3-graphs [29] makes it not far-fetched to suspect that the minimum degree conditions in { , }-quasirandom 3-graphs coincide as well.
Open problems. Are the following true?
• For every d > 1/3 and ε > 0, there exist an integer n 0 and a real > 0 such that the following holds whenever n ≥ n 0 . Let H be an n-vertex -dense 3-graph of density d and satisfying δ 2 (H) ≥ n/3 + εn. Then, H has a tight Hamiltonian cycle. • For every d > 1/2 and ε > 0, there exist an integer n 0 and a real > 0 such that the following holds whenever n ≥ n 0 . Let H be an n-vertex -dense 3-graph of density d and satisfying δ 1 (H) ≥ n 2 /4 + εn. Then, H has a tight Hamilton cycle. • In the two questions above replace -denseness with -denseness.
During the review and revision of this manuscript Araújo, Piga, and Schacht [2] announced to have proved that for every ε > 0 there exists a > 0 such that every sufficiently large ( , 1/4 + ε) -dense 3-graph H satisfying δ(H) ≥ ε n−1 2 , contains a tight Hamilton cycle. Moreover, that the constant 1/4 is optimal. Their result implies our Theorem 1.5 and settles some of the questions appearing above. At the time of writing these lines the full proof of their result was not avialable to us.

O u r a p p roa c h
We employ the so called absorbing path method introduced in [35] and further developed in [36,37]. Roughly speaking, this method reduces the problem of finding a tight Hamilton cycle to that of finding a tight cycle supporting two properties. First, it covers all but ζn vertices for some carefully chosen fixed "small" ζ ∈ (0, 1). Second, it contains a special path referred to as an absorbing-path (rigorously defined below) which has the capability of being rerouted using only those "missing" ζn vertices while keeping its ends unchanged and in this manner absorb, so to speak, all missing vertices rendering a tight Hamilton cycle. Numerous reincarnations of this method now exist in the literature see e.g., [14,27,30,34]. We consequently omit a more rigorous outline of this method and proceed directly to the statement of the so called pillar lemmata underlying this method; these being the so called connecting lemma, absorbing-path lemma, path-cover lemma, and reservoir lemma.
By a k-path we mean a 3-graph P on k vertices and k − 2 edges such that there exists a labelling of V (P ) namely v 1 , . . . , v k such that {v i , v i+1 , v i+2 } ∈ E(P ) for every i ∈ [1, k − 2]. It is said that P connects the pairs {v 1 , v 2 } and {v k−1 , v k−2 }; also referred to as the end-pairs or simply the ends of P . Throughout, the term path is used to mean a tight path.
Roughly speaking, in the absorbing path method, the role of the connecting lemma is, as its name suggests, to connect two disjoint pairs of vertices via a short path. A trivial precondition for such a lemma is that the given pairs that are to be connected both admit some non-trivial codegree. The 3-graphs of Theorem 1.4 come equipped with a minimum codegree assumption which although degenerate will be sufficient in order to establish such a lemma owing to the -denseness of the host 3-graph. The 3-graphs of Theorem 1.5, however, do not support a minimum codegree condition. As a result we will require two separate connecting lemmas; one for each of our main results.
Our connecting lemma fitting for Theorem 1.4 reads as follows. The premise of Theorem 1.5 allows for 3-graphs with pairs of vertices having codegree zero or one that is too modest for our methods to work. Fortunately, regardless of any degree conditions, -dense 3-graphs admit a certain statistical minimum codegree condition in the sense that most pairs of vertices admit a meaningful codegree. This we make precise below in (2.4). Unlike Lemma 1.7 then, the connecting lemma fitting for Theorem 1.5 appeals to this statistical minimum codegree condition, and upon a judicious choice of parameters connects pairs of vertices whose codegree is sufficiently high. It is in this lemma that we encounter the following function g(·). Given reals x, y > 0 satisfying x + y > 1, let g(x, y) := min{x, y, (x + y − 1)/(y + 1)} (1.8) The inequality α + d > 1 appearing in Theorem 1.5 traces back to the third term of this function. Lemma 1.9. (Connecting lemma: 1-degree) For every d 1.9 , α 1.9 , η 1.9 ∈ (0, 1], satisfying α 1.9 + d 1.9 > 1, and η 1.9 < g(α 1.9 , d 1.9 ), there exist an integer n 1.9 := n 1.9 (d 1.9 , α 1.9 , η 1.9 ) and a real 1.9 := 1.9 (d 1.9 , α 1.9 , η 1.9 ) > 0 such that the following holds for all n ≥ n 1.9 and 0 < < 1.9 . Let H be an n-vertex ( , d 1.9 ) -dense 3-graph satisfying δ(H) ≥ α 1.9 n−1 2 and let {x, y} and {x , y } be two disjoint pairs of vertices each having codegree at least (d 1.9 − η 1.9 )n. Then, there exists a 10-path in H connecting {x, y} and {x , y }.
It is in fact true that for -dense 3-graphs, a connecting lemma imposing no minimum degree conditions of any kind is possible. Such a lemma is presented in Lemma 3.23 in § 3.3. Alas, for our needs this lemma is insufficient and this too is explained in § 3.3.
A path A in an n-vertex 3-graph H, is said to be m-absorbing if for every set U ⊆ V (H) \ V (A) with |U | ≤ m there is a path A U having the same ends as A and satisfying V (A U ) = V (A) ∪ U . A path is said to be a (β, µ, κ)-absorbing-path, if it is µn-absorbing, has length at most κn, and both its ends have codegree at least βn in H.
The condition β 1.10 < min{d 1.10 , α 1.10 } appearing in the last lemma seems somewhat puzzling in view that part of the premise of the lemma is that δ 2 (H) ≥ α 1.10 (n − 2). To a certain extent, this condition can be mitigated. We incur it here due to having certain ingredients required for the proofs of Lemma 1.10 and its counterpart, namely Lemma 1.11 stated next, consolidated. This then mandates that β 1.10 < d 1.10 be imposed as to render subsequent applications of (2.4) meaningful. The condition β 1.10 < α 1.10 is admittingly "artificial"; it is kept for brevity purposes seen in the proof of Lemma 1.10.
Given two reals α, β > 0, we write β α to indicate that these can be set such that β, while fixed, can be chosen arbitrarily smaller than α. The aforementioned counterpart of Lemma 1.10 fitting for the setting of Theorem 1.5 reads as follows.
For the next pillar lemma, -denseness is not required. Here, a weaker notion of denseness suffices. Let d, ∈ (0, 1] and let H be an n-vertex 3-graph. If holds for every X ⊆ V (H), then H is said to be ( , d)-dense. If and d are known to exist yet are not made explicit, then we say that H is 1-set-dense 4 . The following lemma imposes no minimum degree conditions on the 3-graph. It will be used in the proofs of both Theorem 1.4 and Theorem 1.5. ) such that the following holds for all n ≥ n 1.13 and 0 < < 1. 13 .
Let H be an n-vertex ( , d 1.13 )-dense 3-graph. Then, all but at most ζ 1.13 n vertices of H can be covered using at most 1.13 vertex-disjoint paths.
The fourth and last pillar lemma is the reservoir lemma. We employ the all encompassing 5 reservoir lemma of [34] which will service both Theorem 1.4 and Theorem 1.5.
Then, for every constant ν 1.14 ∈ (0, 1), there exists an n 1.14 := n 1.14 (ν 1.14 ) such that if n ≥ n 1.14 , then there exists a subset R ⊆ V satisfying Lemma 1.14 is somewhat of an overkill as far as Theorem 1.4 is concerned; indeed, the proof of the latter relies rather weakly only on (R.1) and (R.2). The proof of Theorem 1.5, though, requires the full force of Lemma 1.14, so to speak. In particular, it crucially relies on the n −1/3 -terms seen in Lemma 1.14. A reservoir lemma akin to that seen in [35] indeed suffices for Theorem 1.4. The latter, however, is subsumed by Lemma 1.14 and thus omitted.
Constants. For the most part of this account, we tend to keep track over the raw values of the involved constants. We do, however, appeal on occasion to the notation defined above for constants. § 2

. Pa i r s w i t h p o s i t i v e c o d e g r e e
Let H be a 3-graph, to denote the number of edges {z, u, v} ∈ E(H) for which at least one of the (ordered) pairs (z, u) In this definition G is treated as an undirected graph and indeed in the sequel we shall also write deg H (u, v, G) when G is an undirected graph. We find it more convenient to have the following lemma formulated using undirected graphs.
Lemma 2.1. Let d, α, and be positive reals and let H be a ( , d) -dense n-vertex 3-graph. Let G be a graph on V (H), and let Y ⊆ V (G) satisfy

2)
where k := k(n) is an integer. For an integer ∆ := ∆(n), set Then, holds; the claim now follows upon isolating | B ∆ | in the last inequality.
For an n-vertex ( , d) -dense 3-graph H and a fixed real β > 0, define to consist of all unordered pairs of vertices whose codegree is smaller than βn. An argument akin to setting G to be the complete graph on V (H), Y = V (H), k = n − 1, and ∆ = βn in Lemma 2.1, yields an upper bound on |B β |. To see this, consider so that upon isolating B β we arrive at In particular, ignoring the factor of 1/2, we may write the latter making sense whenever β < d.
A consequence of (2.4), is that the set of edges The spanning subgraph H β ⊆ H induced by E(H) \ E <β then consists only of edges each pair of which has codegree at least βn in H. On its own, H β may admit no meaningful minimum degree condition. It does, however, satisfy Consequently, upon a judicious choice of constants, H β inherits (in the sense of (2.5)) a certain level of -denseness from H. This feature arises in the proof of Theorem 1.5 seen in § 6.2. § 3 . C o n n e c t i n g l e m m a s In this section, we prove Lemmas 1.7 and 1.9. In terms of graphs, our approach, for both these lemmas, can be crudely described as follows. In order to connect two prescribed vertices, a sequence of neighbourhoods, called a cascade, is cultivated; one from each vertex. This, until these neighbourhoods expand so large as to render a certain quasirandomness assumption non-trivial giving rise to numerous "links" between the two sequences of neighbourhoods. Two paths are then traced backwards from a "link" to the two prescribed vertices through the two sequences of neighbourhoods; all the while maintaining vertex-disjointness of the paths thus traced.
3.1 C o n n ec t i n g l e m m a : 2 -d eg ree s e t t i n g In this section, we prove Lemma 1.7 which is the connecting lemma fitting for Theorem 1.4. At the centre of our proof of Lemma 1.7 is the structure of cascades; the next section is dedicated to their definition.

Ca sc ade s
Let n be a sufficiently large integer and let H be an n-vertex 3-graph satisfying δ 2 (H) ≥ βn for some fixed real β ∈ (0, 1] independent of n (and such that βn ≤ n − 2, naturally). Fix x and y to be two vertices in H. Below we define the tuple and refer to it as an {x, y} β -cascade; with cascades being a term borrowed from [35]. All members of the above tuple depend on β as well; we omit this from the notation though. In what follows, each of these members is defined. In broad terms, for every i ∈ [3], N i (x, y) denotes a set of vertices that essentially corresponds to the ith coneighbourhood of the pair {x,y}. The parameters (G i (x, y)) i∈ [3] represent certain graphs between these coneighbourhoods which will facilitate the tracking of 5-paths from N 3 (x, y) all the way (back) to {x, y}.
The assumption δ 2 (H) ≥ βn implies that (3.1) Define G 1 := G 1 (x, y) to be the (bipartite) graph whose vertex set is {y} ∪ N 1 and whose edges are given by the set {yz : z ∈ N 1 }. To define N 2 := N 2 (x, y) and G 2 := G 2 (x, y) we proceed in two step. For the first step, set Define G 2 := G 2 (x, y) to be the graph whose vertex set is N 1 ∪ N 2 and whose edges are given by the set The assumption that δ 2 (H) ≥ βn implies that deg G 2 (z) ≥ βn for every z ∈ N 1 . Then, For the second step towards the definitions of N 2 := N 2 (x, y) and G 2 := G 2 (x, y), we discard members of N 2 whose degree in G 2 into N 1 is "too low" as follows. Set (The choice of log n here is completely arbitrary. Any function ω(n) n growing slowly to ∞ will suffice; this will become clear soon). Setting , we arrive at . This concludes the definitions of N 2 := N 2 (x, y) and G 2 := G 2 (x, y).
We turn to the definition of the set N 3 := N 3 (x, y) and the graph G 3 := G 3 (x, y). To that end, associate an auxiliary graph B w := B w (x, y) with every vertex w ∈ N 2 . In particular, for a fixed vertex w ∈ N 2 , let B w be the graph whose vertex set is V (H) and whose edges are given by the set deg Bw (u) ≥ 20} and let G 3 := G 3 (x, y) be the graph whose vertex set is N 2 ∪ N 3 and whose edge set is given by This completes the definition of an {x, y} β -cascade.
We conclude this section by recording a few useful traits of {x, y} β -cascades that will be called upon in subsequent arguments. Continuing with the notation set thus far, fix w ∈ N 2 . Then, where s denotes the number of vertices u ∈ V (H) satisfying deg Bw (u) < 20. Then, for a sufficiently large n this, in particular, implies that The 5-path is made complete with the fact that {x, y, z } ∈ E(H). Let P denote this path.
It remains to construct a 5-path through the cascade of {x , y } connecting {v, w} and {x , y } in such a way as to not meet any vertex of P . The same argument used for constructing P can be used here as well except for one change. This time around, we require a vertex z ∈ N 2 (x , y ) to play the corresponding role assumed by z above. The vertex z must satisfy z / ∈ {x, y, z , x , y , z, u, v, w} (i.e., it has to avoid z as well). Clearly there is enough freedom to do so. (3.8) , and let {x, y} and {x , y } be two disjoint pairs of vertices in V (H). By Lemma 3.7 it suffices to show that the cascades C α (x, y) and Indeed, all edges in G 3 (x, y) (and thus in F ) are of the form y) and (3.9) follows. Set , k = α 2 n/8, and ∆ := dα 2 n/16 we attain A symmetrical argument applied to C α (x , y ) asserts that the set y )) has size at least α 4 n 2 /2 6 , by (3.9); removing degenerate members (i.e., members of the form (x, x)), we retain at least α 4 n 2 /2 7 non-degenerate members of that Cartesian product. The latter set of non-degenerate pairs, we denote by T . Then, Hence, for a sufficiently large n, we may insist on (many choices) w = z and thus form the required {(x, y), (x , y )}-link. This completes the proof of Lemma 1.7.
3.2 C o n n ec t i n g l e m m a : 1 -d eg ree s e t t i n g In this section, we prove Lemma 1.9 which is the connecting lemma fitting for Theorem 1.5. The definition of cascades, seen at § 3.1.1, fits any 3-graph H satisfying δ 2 (H) = Ω(n). As such the construction of cascades makes no appeal to -denseness. In Lemma 1.9, which is furnished with a minimum 1-degree condition only, cascades, as defined, are not at our disposal (at least not verbatim). To prove Lemma 1.9 then, we put forth a definition of a structure to which we refer as refined cascades. The latter is an adaption of cascades to the setting of Lemma 1.9.
While we do follow closely the definition of cascades when defining their refined counterparts, these two structures are quite different from one another. One crucial manifestation of this difference can be seen through the condition α + d > 1 stated in the premise of Theorem 1.5. This condition is, in fact, incurred through the definition of refined cascades. The construction of latter is then the sole "bottleneck" in our approach preventing us from establishing Conjecture 1.6.
Unlike the case of cascades, the construction of their refined counterparts does make appeals to -denseness of the host 3-graph. Consequently, the definitions of cascades and refined cascades are not consolidated.
(3.11) In what follows, we define the tuple and refer to it as an {x, y} α,d,η -refined-cascade. All members of this tuple depend on α, d, and η as well, yet we omit this from the notation. The members N i and G i , i ∈ [3], assume roles analogous to those assumed by N i and G i , i ∈ [3], in the definition of cascades in § 3.1.1. We proceed with the definition of each of the members of the above tuple. Define Treating B η (see (2.3) for a definition) as a graph, we write to denote the graph complementing B η over V (H). The set {z ∈ V (H) : deg H (z, y) ≥ ηn}, appearing in (3.14), is then the neighbourhood of y in B η . One is now reminded of the following remarkable fact established in [34, Claim 3.1], which in our setting (and owing to η < α as imposed in (3.10)) reads as follows.
then |N 1 | = Ω(n). Rewriting this inequality as we note that the inequality η < a+d−1 1+d imposed in (3.10) implies that (3.16) is satisfied. Moreover, it is here at (3.16) that the condition α + d > 1, imposed in Theorem 1.5, stands out. It follows that (3.17) Define G 1 := G 1 (x, y) to be the (bipartite) graph whose vertex set is given by {y} ∪ N 1 and whose edge set is given by {yz : z ∈ N 1 }.
We proceed to defining N 2 (x, y) and G 2 (x, y). Set and define G 2 := G 2 (x, y) to be the graph whose vertex set is N 1 ∪ N 2 and whose edge set is given by By definition of N 1 (see (3.14)), deg H (y, z) ≥ ηn for every z ∈ N 1 so that deg G 2 (z) ≥ ηn holds for every z ∈ N 1 . Then, All but at most d−η n 2 of the edges of G 2 lie in B η , by (2.4) (with β = η in that equation). Consequently, there exists a subgraph G 2 ⊆ G 2 satisfying e(G 2 ) having the property that E(G 2 ) ∩ B η = ∅. Then, by [10, Proposition 1.2.2], there exists a subgraph G 2 ⊆ G 2 satisfying δ(G 2 ) ≥ ηζ 4 n and this completes the definition of G 2 . We conclude this part of the definition by setting N 2 := V (G 2 ) ∩ N 2 . The property δ(G 2 ) ≥ ηζ 4 n, together with the fact that all edges of G 2 are of the form N 1 × N 2 imply that Next, we define N 3 (x, y) and G 3 (x, y). For w ∈ N 2 , let X w be the graph on V (H) whose edge set is given by and let G 3 := G 3 (x, y) be the graph whose vertex set is N 2 ∪ N 3 and whose edge set is given by Then, holds, where here r denotes the number of vertices u ∈ V (H) satisfying deg Xw (u) < 20; the first inequality is owing to E(G 2 ) ∩ B η = ∅, by definition of G 2 , and the last equality is owing to deg G 3 (w) = n − r, by definition of r. We may then write that where the last inequality is assuming n is sufficiently large. Consequently, This concludes the definition of refined cascades and properties thereof.

3.2.2
Pr o of o f Le mm a 1.9 With the definition of refined cascades complete, our proof of Lemma 1.9 follows closely that seen for Lemma 1.7. Indeed, the machinery of links defined for cascades does carry over to refined cascades essentially verbatim.

A co n n ec t i n g l e m m a w i t h n o m i n i m u m d eg ree co n d i t i o n s
In § 1.2 we mentioned that for -dense graphs, a connecting lemma imposing no minimum degree conditions is possible and that such a lemma is insufficient for our needs. In this section, we make this precise.
Given an n-vertex 3-graph H and a real β > 0, define the sequence H =: H 0 ⊇ H 1 ⊇ H 2 · · · of spanning subgraphs of H as follows.  The proof of Lemma 3.23 is that of Lemma 1.7 essentially verbatim. Let the two β-relevant pairs {x, y} and {x , y } per Lemma 3.23 be given. First, construct the cascades C β (x, y) and C β (x , y ) in H (β) (instead of H) while throughout the construction of these replace every appeal to δ 2 H (β) (which may be zero) with an appeal to δ * 2 H (β) . Indeed, the construction of cascades only requires a sufficiently large minimum codegree for the pairs already captured through the edges of the cascades and in this manner one progresses from one level of the cascade to the next. Second, with these cascades constructed note that these exist in H and thus an ({x, y}, {x , y })-link can be found in H using the very same argument seen in the proof of Lemma 1.7 for that stage.
Unfortunately, we were unable to employ Lemma 3.23 in our account. Indeed, in subsequent arguments the connecting lemmas are used repeatedly in order to connect prescribed pairs of vertices which although admit a relatively large codegree are essentially arbitrary. We were unable to determine whether these pairs are also β-relevant (for an appropriate β). For indeed, a pair is β-relevant if it manages to survive the cleanup procedure, so to speak, giving rise to H (β) . Arbitrary pairs of vertices admitting high codegree in H may of course not survive this process.
We do, however, perceive Lemma 3.23 as being relevant to the pursuit of Conjecture 1.6 and consequently mention it here. In this section, we prove Lemmas 1.10 and 1.11. At the core of these proofs stands the notion of a β-absorber which is a variant of what is often referred to as the natural absorber as far as tight cycles in 3-graphs are concerned. We say β-absorber to mean (β, v)-absorber for some v ∈ V (H).
Our proofs of both absorbing-path lemmas are modelled after the same conceptual three step argument seen in [35]. First, a counting lemma for (β, v)-absorbers with the vertex v prescribed is established; this can be seen in Lemma 4.2. Second, the aforementioned counting lemma is employed in a hypergeometric experiment as to establish the existence of a "small" set F of vertex-disjoint β-absorbers that can absorb any set of vertices that is not too "large"; this can be seen in Lemma 4.8. Third, using the connecting lemmas, namely Lemmas 1.7 and 1.9, we "string", so to speak, the members of F into a single path yielding the required absorbing path. Lemmas 4.2 and 4.8, capturing the first two steps in the above outlined approach, are capable of handling both settings considered in Theorems 1.4 and 1.5. This is due to [32,Remark 1.4] asserting that if an n-vertex 3-graph H admits δ 2 (H) = Ω(n), then δ(H) = Ω(n 2 ).
The third step in the above plan, however, we treat separately across the two aforementioned settings. While the overall scheme of the third step is the same between the two settings, it is here that invocations to the two connecting lemmas are made. The inherent differences between these two lemmas compels (us into having) two separate treatments. In   asserts that e(L v ) ≥ α n−1 2 . Then, for a sufficiently large n Sidorenko's conjecture [12,39] is true for the 2-graph P 4 [3], where by P 4 we mean the path consisting of 3 edges and 4 vertices. Then, for sufficiently large n there are at least homomorphisms of P 4 into L β,v . Consequently (and again assuming n is sufficiently large) there is a collection P of at least α 3 n 4 /2 8 labelled copies of P 4 in L β,v . For an ordered pair (u, w) ∈ V (L β,v ) × V (L β,v ), let P 4 (u, w) denote the number of members of P with the form (x, u, w, y). Set K := α 3 2 /2 10 . Owing to (4.4), Then, Isolating | X|, one arrives at As a result, we attain In preparation for two applications of Lemma 2.1, we define three graphs, namely G 1 , G 2 , and G 3 , edges of which collectively capture the members of P. Lemma 2.1 is then applied to G 1 and G 3 (along with additional parameters defined below); the resultant estimates attained from these two applications of the lemma are then used to analyse G 2 . Set where A (u,w) and B (u,w) are the sets of unordered pairs underlying A (u,w) and B (u,w) , respectively. The graphs G 1 , G 2 , G 3 are not necessarily edge disjoint. Define the sets of vertices For if one of these sets, say A (u,w) , violates this inequality, then in contradiction to (u, w) ∈ Y . Consequently, deg G 1 (u), deg G 3 (w) ≥ α 5 n/2 16 for every u ∈ U and every w ∈ W , respectively. Set  Owing to (4.6), e(G 2 ) ≥ α 3 n 2 /2 17 holds. This fact, together with the estimates seen in (4.7), imply that G 2 admits at least Let (u, w) ∈ U × W be good. At least 8 deg H (u, w, G 1 ) − 1 neighbours a of u in G 1 satisfy a = w. Each such neighbour a of u gives rise to a triple (a, u, w) with the property that au ∈ E(L β,v ) so that deg H (a, u) ≥ βn. The triple (a, u, w) extends to at least 9 deg H (w, u, G 3 ) − 2 quadruples (a, u, w, b) satisfying b / ∈ {a, u} and wb ∈ E(L β,v )) so that deg H (w, b) ≥ βn holds. Any quadruple thus formed defines a (β, v)-absorber.
It follows that for a sufficiently large n, a single good pair (u, w) gives rise to at least Then,

Markov's inequality now implies that
|I| < 7γ 2 n (4.12) holds with positive probability. It follows that an F satisfying (4.10), (4.11), and (4.12) exists. Fix one such F . Define F to be the set of quadruples attained from F by, first, removing all quadruples which do not β-absorb any v and, second, from each intersecting pair of quadruples remove one of the members of that pair. Property (F.1) trivially holds for F. To see that (F.2) holds for F, note that for every v ∈ V (H) With Lemmas 4.2 and 4.8 established, we are ready to prove Lemma 1.10. All that remains is to "string", so to speak, the members of F (from Lemma 4.8) into a single path and prove the absorption capabilities of the resulting path. Given a quadruple (x, y, z, w), we refer to (x, y) and (z, w) as the front and rear end-pair of the quadruple respectively.  To be clear, the definition of µ appeals to that of η 4.8 . The latter requires a value for ϕ 4.8 be set; here, we take ϕ 4.8 = β/20. Set (4.14) Let 0 < < 1.10 be fixed, let n be a sufficiently large integer, and let H be an n-vertex ( , d) -dense 3-graph with δ 2 (H) ≥ α(n − 2). Let F denote the set of β-absorbers, existence of which in H is assured by Lemma 4.8 applied with α 4.8 = α, d 4.8 = d, β 4.8 = β, ϕ 4.8 = β/20, and owing to < 4.8 (d, α, β, β/20), by (4.14). Fix an arbitrary ordering of the members of F, namely F 1 , F 2 , . . . , F r , where r := |F| ≤ βn/20, by (F.1).
In what follows, we prove that a path A of the form exists in H, where here each P i is a 10-path connecting the rear end-pair of F i with the front end-pair of F i+1 ; we use • to denote path concatenations along these pairs. If such a path A were to exist, then it would form a (β, µ, κ)-absorbing path. To see this, observe, first, that |V (A)| = 4r + 6(r − 1) ≤ 10r ≤ βn/2 (4.13) = κn. (4.16) Observe, second, that (F.2) together with a standard greedy argument (see, e.g., [35,Claim 2.6]), assert that such a path A would form a µn-absorbing path. Observe, third, that the ends of such a path A would have codegree at least βn for, indeed, β < α, and δ 2 (H) ≥ α(n − 2) (here we utilise the fact that n is sufficiently large). It remains to establish the existence of the aforementioned path. This we do inductively as follows. Put A 1 := F 1 . Suppose that the (partial) path has been defined for some i ∈ [r − 1]. We define A i+1 as follows. Set where (a, b) is the rear end-pair of F i and (c, d) is the front end-pair of F i+1 . The next two claims verify that Lemma 1.7 can be applied to Owing to κ = β/2 the claim follows. Proof. Owing to β < α, δ 2 (H) ≥ α(n − 2), and n being sufficiently large, we may write  pairs (a, b) and (c, d). The path P i+1 , as defined above, exists and consequently A i+1 as well. This completes the proof of the existence of A and thus concludes the proof of the lemma. holds, where here g(·) is as defined in (1.8). The first constraint being our prerogative, we explain the validity of the second. Owing to ζ κ, it suffices to argue that a choice for κ satisfying 2κ g(α − κ, d) (1.8) = min α − κ, d, α − κ + d − 1 d + 1 exists. The first term in the minimisation entails having to require κ α. The second term imposes κ d. The third, and final, term requires κ α+d−1 2(d+1)+1 . We remark that the condition κ η plays no role in the proof. It is, however, mandated in order to accommodate a subsequent application of Lemma 1.11 in the proof of Theorem 1.5 in § 6.2.
To be clear, the definition of µ entails setting ϕ 4.8 := κ/10. Set Let 0 < < 1.11 be fixed and let H be a ( , d) -dense 3-graph satisfying δ(H) ≥ α 1.11 n−1 2 be given. Let F be a set of (d − ζ)-absorbers, existence of which is assured by Lemma 4.8 applied with α 4.8 = α, d 4.8 = d, β 4.8 = d − ζ, and ϕ 4.8 = κ/10; and also owing to < 4.8 (α, d, d − ζ, κ/10) per (4.23). As in the proof of Lemma 1.10, we seek to establish the existence of a path A of the form (4.17). If such a path A were to exist, then it would form a (β − κ, µ, κ)-absorbing path. Indeed, owing to r := |F| ≤ κn/10 its length would be at most κn, by (4.16); it would be µnabsorbing, by [35,Claim 2.6]; and its ends would have codegree at least (d − η)n as these arise from (d − ζ)-absorbers in F and ζ η. It remains to establish the existence of A. Let A i and V i be as defined in (4.17) and (4.18), respectively. Suffice to prove that the pairs (a, b) and (c, d) (per the definition of V i ) can be connected via a 10-path passing through H[V i ]. This we accomplish using Lemma 1.9. Hence, it remains to prove that for every i ∈ [r], H[V i ] adheres to the premise of that lemma. The following claims verify this.
Starting with the -denseness of H[V i ], note that (4.23), the observation that |V i | ≥ (1 − κ)n, and an argument identical to that seen in (4.20) establish the following.  Proof. Start by observing that where the last term on the r.h.s. accounts for all pairs involving vertices from V (A i ) ∪ V (F). Owing to (4.16) and relaying on n being sufficiently large, we may write Having set κ α + d − 1 in (4.22), implies that (α − κ) + d > 1 holds. This, coupled with the condition ζ + κ g(α − κ, d), also set in (4.22), imply that any two disjoint pairs of vertices having codegree at least (d − ζ − κ)n can be connected in H[V i ] via a 10-path, by Lemma 1.9. In H[V i ], both pairs (a, b) and (c, d) have codegree at least (d − ζ − κ)n and thus connectable in this fashion. The existence of A is established. This concludes the proof of the lemma. § 5 . T h e pat h -c ov e r l e m m a In this section we prove our path-cover lemma, namely Lemma 1.13. Our proof of this lemma employs the weak regularity lemma stated below in Lemma 5.2. In § 5.2, we provide an alternative proof of Lemma 1.13 for graphs equipped with the notion of -denseness; the latter notion is a stronger notion than that of 1-set-denseness assumed in Lemma 1.13. If -denseness is assumed, then the regularity lemma is no longer needed giving rise to a much shorter proof.

P a t h co v e r s i n 1 -s e t -d e n s e 3 -g ra p h s
. A t-partite H is said to be equitable if its t-partition satisfies |V 1 | ≤ |V 2 | ≤ · · · ≤ |V t | ≤ |V 1 | + 1. We also refer to the partition itself as equitable. Let H be an n-vertex 3-graph. Then, there exists an integer t satisfying t 5.2 ≤ t ≤ T 5.2 and an equitable partition V (H) = V 1∪ V 2∪ · · ·∪V t such that for all but at most εt 3 Given a 3-graph H, regularised per Lemma 5.2, and a real d > 0, define R d := R d (H) to denote the 3-graph whose vertices are the clusters (i.e., sets) (V i ) i∈ [t] and whose edges are the triples An edge e ∈ E(H) is said to be crossing with respect to X if there are three clusters V i , V j , V k captured by X such that ) For all 0 < ε < d < 1, every (ε, d)-lowerregular 3-partite equitable 3-graph H on n vertices, with n a sufficiently large integer, contains a family P of vertex disjoint-paths such that for each P ∈ P we have The following is a triviality whose proof is included for completeness.
We are now ready to prove our path-cover lemma, namely Lemma 1.13.
Let n be sufficiently, let < 1.13 , and let H be an n-vertex ( , d)-dense 3-graph. Let R d := R d (H) denote the reduced graph of H obtained after regularising H using the weakregularity lemma, namely Lemma 5.2, applied with ε 5.2 = ε reg and t 5.
Proof. Fix X ⊆ V (R d ) and let C X denote the number of edges of H which are crossing with respect to X and that lie in (ε reg , d )-regular triples H[V i , V j , V k ], where the sets V i , V j , V k are taken from the underlying regularity partition. Then, e R d (X) ≥ C X /2(n/t) 3 ; the factor 2 appearing here is incurred in order to cope with the the fact that cluster sizes are in the set {n/t, n/t + 1}; we use the fact that for a sufficiently large n, 2(n/t) 3 ≥ (n/t + 1) 3 holds.
Observe that Indeed, the second and third terms on the r.h.s. arise from the removal of all edges that have at least two of their vertices in the same cluster captured by X from E(H[∪X]). The fourth term on the r.h.s. arises due the removal of all (crossing) edges found in ε reg -irregular triples of clusters. Finally, the last term on the r.h.s. arises from the removal of all (crossing) edges found in triples of clusters whose edge density is at most d .
As |X| ≤ t, we arrive at As H is ( , d)-dense, holds. Indeed, the term |X| 3 (n/t) 3 accounts only for edges crossing with respect to X while i∈X |V i | 3 accounts also for triples inside clusters captured by X. By (5.5), t + t 2 ≤ 2t 2 ≤ t 3 /4 holds. By (5.6), ≤ /4 holds. We may now write and the claim follows.
In view of Claim 5.7 and the choice of , it follows, by Lemma 5.4, that R d admits a matching M missing at most max{2, ζt/12} vertices of R d . For each edge as to obtain a system of vertex-disjoint paths as described in Lemma 5.3. Let P denote the system of paths thus generated in H over all edges of M . In each H[V i , V j , V k ] corresponding to an edge (V i , V j , V k ) of M , at most 3 εreg(d −εreg) paths are packed. As |M | ≤ T 5.2 (ε reg , t reg )/3 at most 1.13 paths are thus packed.
It remains to argue that the members of P cover all but at most ζn vertices of H. In each As |M | ≤ t/3, at most 12ε reg n vertices of H are missed this way. From the clusters not covered by M at most max{2, ζt/12}·2n/t vertices of H are missed. Overall at most (12ε reg +max{4/t, ζ/2})n vertices of H are missed. Owing to (5.5), 12ε reg ≤ ζ/2 and t ≥ t reg ≥ 8/ζ (so that 4/t ≤ ζ/2); consequently 12ε reg + max{12/t, ζ/2} ≤ ζ as required.

P a t h -co v e r s i n 3 -s e t -d e n s e 3 -g ra p h s
In this section, we provide a significantly shorter proof for Lemma 1.13, under the strengthened assumption that the host 3-graph is -dense and not merely 1-set-dense. The main ingredient, so to speak, of the argument for constructing path-covers in -dense graphs is the following. Proof of Lemma 1.13 for -dense 3-graphs. Given d := d 1.13 and ζ := ζ 1.13 , set 1.13 := dζ 3 2·3 3 , and set 1.13 := 1/ 1.13 . Let 0 < < 1.13 be fixed, let n be sufficiently large, and let H be a ( , d) -dense n-vertex 3-graph.
We prepare for an application of Lemma 1.14. For each pair of vertices {x, y} Proof. Fix G 1 , G 2 ⊆ V (H ) × V (H ) and note that as H is an induced subgraph of H, then As -denseness implies 1-set-denseness, it follows that H is (ξ 1 , d)-dense for some ξ 1 < 1.13 (d, ζ). Lemma 1.13 then asserts that H admits a collection P := {P 1 , . . . , P h−1 }, h − 1 ≤ 1.13 (d, ζ), of vertex-disjoint paths covering all but at most ζn ≤ ζn vertices of H and thus of H as well. Write P h := A and set P : In what follows we use the set R in order to concatenate the members of P into a (tight) cycle. This entails h applications of the connecting lemma fitting to this setting, namely Lemma 1.7. We proceed in two steps. First, a path of the form is constructed, where here each K i is a 10-path disjoint of all other 10-paths involved in the construction. Second, the remaining "free" end-pair of P h is connected using an additional 10-path with the remaining "free" end-pair of P 1 . Also here we The resulting cycle we denote by C. We now make this precise. The construction of L is done inductively. Set L 1 := P 1 . Assuming has been constructed for some i ∈ [h − 1], we define L i+1 as follows. Let {a, b} be the free end-pair of P i and let {c, d} be one of the end pairs of P i+1 . Set Observing that |V (L i ) ∩ R| ≤ 10h, we may write that holds for a sufficiently large n. Owing to < 1.7 (d, β/8) · (ν(1 − κ)/4) 3 , by (6.2), an argument identical to that seen in Claim 6.6 establishes that H[R i ] is (ξ 2 , d) -dense for some ξ 2 < 1.7 (d, β/4). Moreover, Completing L into the aforementioned cycle C is done using the exact same argument provided for K i .
Writing η = d − (d − η) and appealing to both (2.4) and (6.16), we may write As H d−η spans H , |V (H d−η )| = n holds, and the claim follows. As in the proof of Theorem 1.4, we seek to construct a path L of the form (6.7) and then close the latter into a (tight) cycle C. With the approach for this construction here conceptually identical to that seen in the proof of Theorem 1.4, we focus on the differences. More specifically, we are to show that h applications of the connecting lemma relevant to the setting at hand, namely Lemma 1.9, can be carried out as to construct the cycle C. To that end, for i ∈ [h − 1], define L i and R i as in (6.8) and (6.9), respectively.
Next, we consider the codegree of the pairs {a, b} and {c, d}, per the definition of R i .
Next, we address the -denseness of H[R i ].
This completes the definition of the cycle C containing the absorbing-path A. To prove that C can be extended as to absorb all possibly uncovered vertices, use the argument seen for this in the proof of Theorem 1.4. This concludes our proof of Theorem 1.5.