The following states that the minimum vertex degree in terms of codegrees.
Fact 3.12. Let H be a k-partite k-graph with parts of size n such that δ[k]\{i} ≥ai for each
i∈[k], then any i∈[k] and v ∈Vi, we have deg(v)≥maxj6=iajnk−2.
Givenβ >0,i∈N, j ∈[k] and two verticesu, v ∈Vj, we say thatu, v are (β, i)-reachable
in H if and only if there are at leastβnik−1 (ik−1)-sets W such that bothH[{u} ∪W] and
H[{v} ∪W] contain perfect matchings. W is called reachable set for u, v. If allu, v ∈Vj are
(β, i)-reachable, then we say Vj is (β, i)-closed. Denote by ˜Nβ,i(v) the set of vertices that are
We show that the number of one-step reachable neighbors to any vertex in each part Vi
is not much less than the corresponding codegree ai, where i∈[k].
Proposition 3.13. Suppose0<1/n α 1/k and letH be a k-partitek-graph such
that δ[k]\{1}(H), δ[k]\{2}(H)≥n. For any j ∈[k] and v ∈Vj, |N˜α,1(v)| ≥δ[k]\{j}(H)−
√
αn.
Proof. Fix a vertexv ∈Vj for somej ∈[k], note that for any other vertexu∈Vj,u∈N˜α,1(v)
if and only if |NH(u)∩NH(v)| ≥αnk−1. By double counting, we have
X
S∈NH(v)
degH(S, Vj)<|N˜α,1(v)| · |NH(v)|+n·αnk−1.
For any S in the above inequality, we know that degH(S, Vj)≥δ[k]\{j}(H). Moreover, since
v is not in one of V1 and V2, we have that
|NH(v)| ≥nk−2n≥ √ αnk−1, as α. Thus, |N˜α,1(v)|> δ[k]\{j}(H)− αn k |NH(v)| ≥δ[k]\{j}(H)− √ αn as desired.
Throughout the rest of this subsection, without loss of generality, we may assume only
a1, a2 ≥ n. The following is the key point to our proof. Here we only give a tentative
outline.
Lemma 3.14 (draft). Given 0 < 0, γ , ∗ and sufficiently large n, there exists α > 0
such that the following holds. Let H be a k-partite k-graph with parts of size n such that δ[k]\{i} ≥ ai for each i ∈ [k]. If
P
i∈[`]ai ≥ (1−γ)n, a1 ≥ a2 ≥ n and aj < n for j ≥ 3,
then one of the following holds.
(i) a1 ≥a2 ≥n/2−kn, H is ∗-D-extremal.
(ii) There exists a matching M0 of size |M0| ≤0n such that for every legal k-set S of H, the number of S-perfect-absorbing sets in M0 is at least αn.
Proof. Given 0< 0 and sufficiently large n, let H be a k-partitek-graph with parts of size n such thatδ[k]\{1}, δ[k]\{2} ≥n. By Fact 3.12,
δ10(H)≥min{a1nk−2, a2nk−2} ≥nk−1.
Claim 3.15. If any of Vj where j ∈ [2] is β-closed for some β > 0, then there exists a
matching M0 in H of size |M0| ≤ 0n and α > 0 such that for every legal k-set S of H, the number of S-perfect-absorbing sets in M0 is at least αn.
Proof. If one ofV1, V2isβ-closed for someβ >0, assumeV1is (β, i0)-closed, i.e., anyu, v ∈V1
are (β, i0)-reachable.
Fix a legal k-set S = {v1, v2, . . . , vk} such that vj ∈ Vj, we claim there are at least
βni0k/2 S-perfect-absorbing i
0k-sets. First of all, we find v01 ∈ V1 \ {v1} such that
{v10, v2, . . . , vk} spans an edge. Since deg(S \ {v1}) ≥ n, there are at least n−1 choic-
es ofv10. Since V1 isβ-closed, there are at least βni0k−1 reachable (i0k−1)-setsW forv1 and v10. Among them, at leastβni0k−1−(k−1)nk−2 ≥βni0k−1/2 reachable (i
0k−1)-sets W are
disjoint from S. In total, we have at leastβni0k/2 S-perfect-absorbing sets. Next we build
the matching M0 by applying Proposition 3.14.
We have two cases.
Case 1: If a1 ≥ n/2 +n, then V2 is (2,1)-closed. Indeed, by Fact 3.12, any v ∈ V2
has deg(v) ≥ (1/2 +)nk−1, therefore, for any u, v ∈V
2, we have|N1(u)∩N1(v)| ≥ 2nk−1.
By Claim 3.15, (ii) is true.
Case 2: Ifa1 < n/2+n, sincea1+a2 ≥n−γn−(k−2)n, we havea1 ≥a2 > n/2−kn.
In this case, for any v ∈V, by Fact 3.12, deg(v)≥(1/2−k)nk−1.
Claim 3.16. For any i ∈ [k], either Vi is β-closed for some β > 0 or there is a partition
Vi =Xi0∪Y 0 i such that X 0 i and Y 0 i are (β
0,1)-closed for some β0 >0.
Proof. Fix i ∈ [k]. If for any pair of vertices xi, yi ∈ Vi , there exists α > 0 such that
and |N(yi)∩N(z)| ≥αnk−1, then Vi is (β,2)-closed for some β >0.
We may assume that there existsxi, yi ∈Vi such that for any α >0,|N(xi)∩N(yi)|<
αnk−1and, at mostαnverticesz ∈Visuch that|N(xi)∩N(z)| ≥αnk−1 and|N(yi)∩N(z)| ≥
αnk−1. In this case, let X
i = {v ∈ Vi : |N(yi)∩ N(v)| < αnk−1} and Y = {v ∈ Vi :
N(xi)∩N(v)|< αnk−1}. Let Zi =Vi\(Xi∪Yi). We have the following properties of Xi, Yi
and Zi.
(i)xi ∈Xi and yi ∈Yi by definations of Xi, Yi.
(ii) Xi∩Yi =∅. Supposev ∈Xi∩Yi. |N(xi)∪N(yi)∪N(v)|=|N(v)\N(xi)∪N(yi)|+|N(xi)\N(yi)|+|N(yi)| >3(1 2 −)n k−1−3αnk−1 > nk−1, a contradiction. (iii) |Zi|< αn
(iv) For anyx, x0 ∈Xi,|N(x)∆N(x0)|<8αnk−1, and hencex, x0 are 1-reachable to each
other. The same holds for any pair of vertices in Yi.
(v) For anyx∈Xi and y∈Yi, |N(x)∩N(y)|<5αnk−1
For vertexz ∈Zi, if zi is 1-reachable to any vertexx∈Xi, then addz toXi; otherwise,
there existsx0 ∈Xi such that |N(x0)∩N(z)|< nk−1. In the later case, we claim that z is
1-reachable to anyy∈Yi, and hence we addz toYi. Fory∈Yi. assume|N(y)∩N(z)|< 0n,
then |N(x0)∪N(y)∪N(z)|> nk−1, a contradiction. Denote the resulted sets as Xi0 and Y
0
i
, which will be the desired partition.
By Proposition 3.13 with α , for i = 1,2, |N˜α,1(v)| ≥ δ[k]\{i}(H)−
√
αn > (1/2−
k−√α)n . So |Xi0|,|Yi|>(1/2−0)n fori= 1,2.
After having the partition of each part, we need to consider the edge set of H. This is
the hard part, and more work need to be done.
give an outline to solve the non-extremal case. For the extremal case, we need to handle two subcases: the space barrier similar to the one we did and the divisibility barrier.