Extremal Examples

In this section, we prove the lower bound in Theorem 4.1 by six constructions. Following

the definition in [70], we say a 3-partite 3-graph Ka,b,c is of type 0 if gcd(a, b, c) > 1 or

a=b=c= 1. We say Ka,b,c is of type d≥1 if gcd(a, b, c) = 1 and gcd(b−a, c−b) =d.

Construction 4.4 (Space Barrier I). Let V1 and V2 be two disjoint sets of vertices such that

|V1|= a_kn−1and|V1|+|V2|=n. LetG1 be the3-graph onV1∪V2whose edge set consists of all triples esuch that|e∩V1| ≥1. Thenδ1(G1) = n−₂1

− (1−ak)n

2

= (1−(1−a/k)2) ₂n+o(n2_). Since a≤b≤c, we have a≤k/3 and 0<1−(1−a/k)2 ≤5/9.

We claim thatG1 has no perfectKa,b,c-tiling. Indeed, consider a copyK0 ofKa,b,c inG1.

We observe that at least one color class ofK0 is a subset ofV1 – otherwiseV2 contains at least

one vertex from each color class; since K0 is complete, there is an edge in V2, contradicting

the definition ofG1. Hence aKa,b,c-tiling in G1 covers at most |V_a1|k < nvertices, so it cannot

Figure 4.1. Space barriers

Construction 4.5(Space Barrier II). LetV1 andV2 be two disjoint sets of vertices such that

|V1|= a+_kbn−1and |V1|+|V2|=n. Let G2 be the 3-graph on V1∪V2 whose edge set consists of all triples e such that |e∩V1| ≥ 2. Then δ1(G2) =

a+b k n−1 2 = ((a+b)2_/k2₎ n 2 +o(n2_). Since a≤b≤c, we have a+b≤2k/3 and 0<(a+b)2_/k2 _≤4/9.

We claim that G2 has no perfect Ka,b,c-tiling. Similarly as in the previous case, for any

copyK0 ofKa,b,c inG2, at least two color classes ofK0 are subsets ofV1. Hence a Ka,b,c-tiling

inG2 covers at most _a|V₊1_b|k < nvertices, so it cannot be perfect.

Construction 4.6(Divisibility Barrier I). LetV1 andV2 be two disjoint sets of vertices such that |V1|= bn₂c+ 1 and |V1|+|V2| =n. Let H1 be the 3-graph on V1∪V2 such that H1[V1] and H1[V2] are two complete 3-graphs. Then δ1(H1)≥

n 2−2 2 = 1 4 n 2 +o(n2_).

We claim that H1 has no perfect Ka,b,c-tiling. Indeed, each copy of Ka,b,c must be

a subgraph of H1[V1] or H1[V2]. Since k ≥ 3, due to the choice of V1 and V2, we have

|V1| 6=|V2| modk and therefore k cannot divide both |V1|and |V2|. Hence H1 has no perfect

Ka,b,c-tiling.

Construction 4.7 (Divisibility Barrier II). Suppose that Ka,b,c is of typedfor some evend.

LetV1 andV2 be two disjoint sets of vertices such that|V1|+|V2|=n and|V2| ∈[n₂−2,n₂+ 2] is odd, and gcd(a, b, c) _- |V2| if gcd(a, b, c) > 1. Note that we can pick |V2| satisfying these conditions because in the interval[n₂−2,n₂+2], there are at least two consecutive odd numbers,

Figure 4.2. Divisibility barriers

therefore at least one of them is not divisible by gcd(a, b, c). Let H2 be the 3-graph on V1 ∪ V2 whose edge set consists of all triples e such that |e∩V2| is even (0 or 2). Then δ1(H2) = min{ |V1|−1 2 + |V2| 2 ,|V1|(|V2| −1)}= 1₂ n₂ +o(n2).

We claim that H2 has no perfectKa,b,c-tiling. Consider a copyK0 of Ka,b,c inH2. Since

every edge intersects V2 in an even number of vertices andK0 is complete, no color class of

K0 intersects both V1 and V2. Moreover, either 0 or 2 color classes of K0 are subsets of V2.

Thus |V(K0)∩V2| ∈ {0, a+b, a+c, b+c}. If gcd(a, b, c)>1, then |V(K0)∩V2| is divisible

by gcd(a, b, c). Since gcd(a, b, c) _- |V2|, there is no perfect Ka,b,c-tiling. Otherwise, either

a=b=c= 1 or gcd(b−a, c−b) is even. In either case, all of a+b, a+cand b+care even

and thus |V(K0)∩V2| is even. Since |V2| is odd, H2 has no perfect Ka,b,c-tiling.

Construction 4.8 (Divisibility Barrier III). Suppose that Ka,b,c is of type d for some odd

d ≥ 3, let V1 and V2 be two disjoint sets of vertices such that |V1|+|V2| = n and |V1| ∈

[n₃ −1,n₃ + 1] and d_-(|V1| −n_ka). Let H3 be the 3-graph on V1∪V2 whose edge set consists of all triples e such that |e∩V1|= 1. Then δ1(H3) = min{|V1|(|V2| −1),

|V2|

2

}= 4₉ n₂+o(n2_).

We claim that H3 has no perfect Ka,b,c-tiling. Consider a copy K0 of Ka,b,c in H3.

Similarly as in the previous case, exactly one color class of K0 is a subset of V1, which

implies |V1 ∩V(K0)| ∈ {a, b, c}. Since gcd(b−a, c−b) = d, we have a ≡ b ≡ c mod d and

Figure 4.3. Tiling barrier

|V(K)∩V1| −n_ka≡0 modd, contradicting our assumption on|V1|. HenceH3 has no perfect

Ka,b,c-tiling.

Construction 4.9 (Tiling Barrier). Let α =√2−1 and suppose that V is partitioned into

{v} ∪V1∪V2∪V3 such that |V1|=|V2|=αn and |V|=n. Define a 3-graph F on V whose edge set consists of all triples vxy with x ∈V1, y ∈V2 and all triples e in V1 ∪V2∪V3 such that e∩V1 =∅ or e∩V2 =∅. Therefore, δ1(F) = (6−4

√

2) n₂+o(n2)≈0.343 n₂.

It is easy to see that v is not contained in any copy of K2,2,2, and hence not contained

in any copy of Ka,b,c with a >1. Therefore, F has no perfect Ka,b,c-tiling with a >1.

Proof of Theorem 4.1 (lower bound). Given positive integers a ≤ b ≤ c and n ∈ k_N, where

k =a+b+c, let t1 =t1(n, Ka,b,c) be the tiling threshold. By Constructions 4.4 and 4.5, we

havet1 ≥(1−(1−a/k)2) n₂ +o(n2_{) andt} 1 ≥((a+b)2/k2) n₂ +o(n2_{). Furthermore, assume}

Ka,b,chas type d. First, by definition,dis even if and only if gcd(a, b, c)>1 ora=b=c= 1

ord≥2 is even, in this case by Construction 4.7, we havet1 ≥ 1₂ n₂

+o(n2_{). Second, assume}

that d ≥ 3 is odd, then by Construction 4.8, we have t1 ≥ 4₉ n₂

+o(n2_{). Finally assume}

that d = 1. If a = 1, by Construction 4.6, we have t1 ≥ 1₄ n₂

+o(n2). If a ≥ 2, then by

Construction 4.9, we have t1 ≥(6−4

√