The Connecting and Absorbing Lemmas

Since pěn^´1{p<sup>k`r´2</sup>k´1 q<sup>´ε</sup>, and `_i`1

k

˘ε ď1{3 for any k´1ďiďk`r´2, if iě2, then

n^{i`1</sup>pp<sup>i`1}k q ěn^{i`1</sup>n<sup>´p1{</sup>p<sup>k`r´2}k´1 q^`εqi`1k pk´1ⁱ q ěn^{i`1´i`1k ´}p^i`1k q^ε ěCn.

Otherwise i “ 1 and thus k “2, in which case we have n^{i`1</sup>pp<sup>i`1}k q “ n²p ě Cn.

Therefore,

min

0ăeHďgpk`r´1qn^vHp^eH ěCn. (4.7)

From (4.6) and (4.7), we have Φ_P^r

b ěCn, as desired.

with twh, . . . , w1u and there are at least pα¹n{2q^h choices for the ordered set pv_{h`1</sub>, . . . , v<sub>2h</sub>q. Similarly, we can extend S<sup>1</sup> to an pr, kq-path pw<sub>2h</sub>, . . . , w<sub>1</sub>q such that the vertices of this pr, kq-path are disjoint with tv<sub>1</sub>, . . . , v<sub>2h</sub>u and there are at least pα<sup>1</sup>n{2q<sup>h</sup> choices for the ordered set pw<sub>2h</sub>, . . . , w<sub>h`1}q.

Therefore there are at least pα¹n{2q^2h ě24βn^2h possible choices for the ordered 2h-sets pv_{h`1</sub>, . . . , v<sub>2h</sub>, w<sub>2h</sub>, . . . , w<sub>h`1}q. Let C_S,S¹ be a collection of ex-actly 24βn^2h such ordered 2h-sets of vertices. Clearly if an ordered set C inC_S,S¹ spans a copy of P_2h^r , then C connects S and S¹.

Now we will use the edges ofG“G^pkqpn, pq to obtain the desired copies ofP_2h^r that connect the pairs in S. Let T be the set of all labelled copies of P_2h^r in G.

We claim that the following properties hold with probability at least 1´3{? n:

(a) |T| ď2p^tn^2h;

(b) for everytS, S¹u PS, at least 12βp^tn^2h members of T connectS and S¹; (c) the number of overlapping pairs of members of T is at most 4p2hq²p^2tn^4h´1.

To see that the claim above holds, note that by Proposition 64, we can apply Lemma 63with F “P_2h^r , γ “24β and C_S,S¹ in place of F_i. Items (a),(b) and (c) follow, respectively, from Lemma 63 (iii), (ii) and (iv).

Next we select a random collection C¹ by including each member of T in-dependently with probability q :“ β{p2p2hq²n^2h´1p^tq. We remark that q ă 1, since n^2h´1p^těC due to Proposition 64. By using Chernoff’s inequality (for (i) and (ii) below) and Markov’s inequality (for (iii) below), we know that there is a choice of C¹ that satisfies the following properties:

(i) |C¹| ď2q|T| ďβn;

(ii) for everytS, S¹u PS, there are at least 12βpq{2qn^2hp^t“3β²n{p2hq² members of C¹ that connectS and S¹;

(iii) the number of overlapping pairs of members of C¹ is at most 8p2hq²q²n^4h´1p^2t “2β²n{p2hq².

Deleting one member from each overlapping pair, we obtain a collectionC of vertex disjoint copies of P_2h^r with |C| ďβn, and such that, for every pair of disjoint or-dered h-sets each spanning a K_h^pkq in H, there are at least 3β²n{p2hq²´2β²n{p2hq² “β²n{p2hq² sets of 2hvertices connecting them.

4.2.2 The Absorbing Lemma

In this subsection we prove our absorbing lemma.

Lemma 66 (Absorbing Lemma). Suppose 1{n!ε!ζ !α!1{k,1{r. Let H be an n-vertex k-graph with δ_k´1pHq ě p1´c`αqn and suppose p“ppnq ěn^´c´ε. Then a.a.s. H YG^pkqpn, pq contains an pr, kq-path P_abs of order at most 6hζn such that, for every set X Ď VpHqrVpP_absq with |X| ď ζ²n{p2hq², there is an pr, kq-path inH on VpPabsq YX that has the same ends as Pabs.

We call the pr, kq-path P_abs in Lemma 66 an absorbing path. We now define absorbers.

Definition 67. Let v be a vertex of a k-graph. An ordered 2h-set of vertices pw₁, . . . , w_2hq is a v-absorber if pw₁, . . . , w_2hq spans a labelled copy of P_2h^r and pw₁, . . . , w_h, v, w_{h`1</sub>, . . . , w<sub>2h</sub>q spans a labelled copy of P<sub>2h`1}^r .

Proof of Lemma 66. Suppose 1{n!ε!ζ !β !α!1{k, 1{r. We split the proof into two parts. We first find a setF of absorbers and then connect them to anpr, kq-path by using Lemma 65 (Connecting Lemma). We will expose G “ G^pkqpn, pq in two rounds: G “ G₁YG₂ with G₁ and G₂ independent copies of G^pkqpn, p¹q, where p1´p¹q² “1´p.

Fix a vertex v. By the codegree condition of H, we can extend v to a labelled copy of P_{2h`1</sub><sup>r</sup> in the form pw<sub>1</sub>, . . . , w<sub>h</sub>, v, w<sub>h`1}, . . . , w_2hq such that there are at least pαn{2q^2h ě24ζn^2h choices for the ordered 2h-set pw₁, . . . , w_2hq. Let A_v be a collection of exactly 24ζn^2h such ordered 2h-sets. By definition, if an ordered setA in A_v spans a labelled copy ofP_2h^r , thenA is a v-absorber.

Now considerG₁ “G^pkqpn, p¹q and letT be the set of all labelled copies ofP_2h^r in G₁. By Proposition 64, we can apply Lemma63 with F “P_2h^r and A_v in place of F_i. Using the union bound we conclude that the following properties hold with probability at least 1´3{?

n:

(a) |T| ď2p^tn^2h;

(b) for every vertexv in H, at least 12ζp^tn^2h members of T are v-absorbers;

(c) the number of overlapping pairs of members of T is at most 4p2hq²p^2tn^4h´1.

Next we select a random collection F¹ by including each member of T indepen-dently with probabilityq“ζ{p2p2hq²p^tn^2h´1q ă 1. In view of the properties above, by using Chernoff’s inequality (for (i) and (ii) below) and Markov’s inequality (for (iii) below), we know that there is a choice of F¹ that satisfies the following

properties:

(i) |F¹| ďζn;

(ii) for every vertex v, at least 12ζpq{2qp^tn^2h “ 3ζ²n{p2hq² members of F¹ are v-absorbers;

(iii) there are at most 8p2hq²q²n^4h´1p^2t “2ζ²n{p2hq² overlapping pairs of mem-bers of F¹.

By deleting from F¹ one member from each overlapping pair and all members that are not in T, we obtain a collection F of vertex-disjoint copies of P_2h^r such that |F| ďζn, and for every vertex v, there are at least

3ζ²n{p2hq²´2ζ²n{p2hq² “ζ²n{p2hq² v-absorbers.

Now we connect these absorbers using Lemma 65. Let V¹ “ VpHqrVpFq andn¹ “ |V¹|. In particular,n¹ ěn{2 is sufficiently large. Now considerH¹ “HrV¹s and G¹ “ G₂rV¹s “ G^pkqpn¹, p¹q. Since |VpFq| ď 2h ¨ ζn ď α²n, we haveδ_k´1pH¹q ě p1´c`α{2qn. We apply Lemma65to H¹ andG¹ with α¹ “α{2 and β, and conclude that a.a.s. H¹YG¹ contains a set C of vertex-disjoint copies of P_2h^r such that|C| ďβn and for every pair of ordered h-sets in V¹, there are at least β²n members of C connecting them.

For each copy of P_2h^r in F, we greedily extend its two ends by h vertices such that all new paths are pairwise vertex disjoint and also vertex disjoint from VpCq. This is possible because of the codegree condition of H0 and the fact that |VpFq| `2h|F| ` |VpCq| ď 2hζn`2hζn`2h¨βn ă αn{4. Note that both ends of these pr, kq-paths P_4h^r are in V¹ rVpCq. Since ζn ď β²n¹{p2hq², we can greedily connect these P_4h^r . Let P_abs be the resulting pr, kq-path. By construction,|VpPabsq| ď p4h`2hq¨ζn“6hζn. Moreover, for anyX ĎVrVpPabsq such that |X| ďζ²n{p2hq², since each vertex v has at least ζ²n{p2hq² v-absorbers in F, we can absorb them greedily and conclude that there is an pr, kq-path onVpP_absq YX that has the same ends as P_abs.