Background, Graph Packing ◮ Corollary: (B. Bollob´ as, S. E. Eldridge [1976]) Suppose that G and H are graphs with n vertices each. ◮ Suppose that ∆( G ) = n − 1 and δ ( H ) ≥ 1 or δ ( G ) ≥ 1 and ∆( H ) = n − 1 (*) ◮ If (*) holds then there is no packing of G and H . ◮ If (*) does not hold, ( G , H ) is not one of the pairs in the two figures and | E ( G ) | + | E ( H ) | ≤ 2 n − 3, then there is a packing G and H . ◮ If (*) does not hold and | E ( G ) | + | E ( H ) | ≤ 2 n − 4, then there is a packing G and H .
Background, Graph Packing ◮ Corollary: (B. Bollob´ as, S. E. Eldridge [1976]) Suppose that G and H are graphs with n vertices each. ◮ Suppose that ∆( G ) = n − 1 and δ ( H ) ≥ 1 or δ ( G ) ≥ 1 and ∆( H ) = n − 1 (*) ◮ If (*) holds then there is no packing of G and H . ◮ If (*) does not hold, ( G , H ) is not one of the pairs in the two figures and | E ( G ) | + | E ( H ) | ≤ 2 n − 3, then there is a packing G and H . ◮ If (*) does not hold and | E ( G ) | + | E ( H ) | ≤ 2 n − 4, then there is a packing G and H . ◮ If | E ( G ) | + | E ( H ) | < 3 n − 2 , then G and H pack. 2
Background, Graph Packing ◮ Corollary yields that B. Bollob´ as, S. E. Eldridge’s Theorem can be restated as follows.
Background, Graph Packing ◮ Corollary yields that B. Bollob´ as, S. E. Eldridge’s Theorem can be restated as follows. ◮ Theorem: Let G and H be n -vertex graphs with | E ( G ) | + | E ( H ) | ≤ 2 n − 3. Then G and H do not pack if and only if either ( G , H ) is one of the seven pairs in two figures above, or one of G and H has a universal vertex and the other has no isolated vertices.
Background, Graph Packing ◮ Corollary yields that B. Bollob´ as, S. E. Eldridge’s Theorem can be restated as follows. ◮ Theorem: Let G and H be n -vertex graphs with | E ( G ) | + | E ( H ) | ≤ 2 n − 3. Then G and H do not pack if and only if either ( G , H ) is one of the seven pairs in two figures above, or one of G and H has a universal vertex and the other has no isolated vertices. ◮ To see that B. Bollob´ as, S. E. Eldridge’s Theorem yields N. Sauer, J. Spencer’s Theorem, observe that for each pair ( G , H ) in the above two figures, | E ( G ) | + | E ( H ) | = 2 n − 3 ≥ 1 . 5 n − 1 and that if G has a universal vertex and H has isolated vertices, then | E ( G ) | + | E ( H ) | ≥ ( n − 1) + ⌈ n / 2 ⌉ .
Background, Hypergraph Packing ◮ Edges of size 1, n − 1 or n make harder for hypergraphs to pack.
Background, Hypergraph Packing ◮ Edges of size 1, n − 1 or n make harder for hypergraphs to pack. ◮ For example, if V ( G ) is an edge in G and V ( H ) is an edge in H , then G and H do not pack.
Background, Hypergraph Packing ◮ Edges of size 1, n − 1 or n make harder for hypergraphs to pack. ◮ For example, if V ( G ) is an edge in G and V ( H ) is an edge in H , then G and H do not pack. ◮ If the total number of 1-edges or the total number of ( n − 1)-edges in G and H is at least n + 1, then G and H also do not pack.
Background, Hypergraph Packing ◮ Theorem: (M. Pil´ sniak and M. Wo´ zniak [2007]) If an n -vertex hypergraph G has at most n / 2 edges and V ( G ) is not an edge in G , then G packs with itself.
Background, Hypergraph Packing ◮ Theorem: (M. Pil´ sniak and M. Wo´ zniak [2007]) If an n -vertex hypergraph G has at most n / 2 edges and V ( G ) is not an edge in G , then G packs with itself. ◮ They asked whether such G packs with any n -vertex hypergraph H satisfying the same conditions.
Background, Hypergraph Packing ◮ Theorem: (M. Pil´ sniak and M. Wo´ zniak [2007]) If an n -vertex hypergraph G has at most n / 2 edges and V ( G ) is not an edge in G , then G packs with itself. ◮ They asked whether such G packs with any n -vertex hypergraph H satisfying the same conditions. ◮ Theorem: (P. Naroski [2009]) Let G and H be n -vertex hypergraphs with no n -edges. If | E ( G ) | + | E ( H ) | ≤ n , then G and H pack. ◮ By the above examples, the bound of n in Naroski’s Theorem is sharp.
Background, Hypergraph Packing ◮ Theorem: (M. Pil´ sniak and M. Wo´ zniak [2007]) If an n -vertex hypergraph G has at most n / 2 edges and V ( G ) is not an edge in G , then G packs with itself. ◮ They asked whether such G packs with any n -vertex hypergraph H satisfying the same conditions. ◮ Theorem: (P. Naroski [2009]) Let G and H be n -vertex hypergraphs with no n -edges. If | E ( G ) | + | E ( H ) | ≤ n , then G and H pack. ◮ By the above examples, the bound of n in Naroski’s Theorem is sharp. ◮ Theorem: (Naroski [2009]) Let G and H be n -vertex hypergraphs without edges of size smaller than k and greater than n − k for some 1 ≤ k ≤ ⌊ n 2 ⌋ such that � n � | E ( G ) || E ( H ) | < . Then G and H pack. k
Our Main Result, Hypergraph Packing ◮ Definition: A bad pair of hypergraphs to be either one of the bad pairs ( G ( i ), H ( i )) of B. Bollob´ as, S. E. Eldridge, or the pair of hypergraphs obtained from some G ( i ) and H ( i ) by replacing each of the graph edges by its complementary edge.
Our Main Result, Hypergraph Packing ◮ Definition: A bad pair of hypergraphs to be either one of the bad pairs ( G ( i ), H ( i )) of B. Bollob´ as, S. E. Eldridge, or the pair of hypergraphs obtained from some G ( i ) and H ( i ) by replacing each of the graph edges by its complementary edge. ◮ An edge e ′ in an n -vertex hypergraph F is complementary to edge e if e ′ = V ( F ) − e .
Our Main Result, Hypergraph Packing ◮ Definition: A bad pair of hypergraphs to be either one of the bad pairs ( G ( i ), H ( i )) of B. Bollob´ as, S. E. Eldridge, or the pair of hypergraphs obtained from some G ( i ) and H ( i ) by replacing each of the graph edges by its complementary edge. ◮ An edge e ′ in an n -vertex hypergraph F is complementary to edge e if e ′ = V ( F ) − e . ◮ Theorem: (H., Kostochka, Stocker [2011]). Let G and H be n -vertex hypergraphs with | E ( G ) | + | E ( H ) | ≤ 2 n − 3 containing no 1-edges and no edges of size at least ( n − 1). Then G and H do not pack if and only if either ( G , H ) is a bad pair or one of G and H has a universal vertex and every vertex of the other is incident to a graph edge.
Hypergraph Packing ◮ Since each of the graphs in the above two figures has at most 9 vertices, for n ≥ 10 the theorem says that: Let G and H be n-vertex hypergraphs with | E ( G ) | + | E ( H ) | ≤ 2 n − 3 containing no 1 -edges and no edges of size at least ( n − 1) , then G and H do not pack if and only if one of G and H has a universal vertex and every vertex of the other is incident to a graph edge.
Hypergraph Packing ◮ Since each of the graphs in the above two figures has at most 9 vertices, for n ≥ 10 the theorem says that: Let G and H be n-vertex hypergraphs with | E ( G ) | + | E ( H ) | ≤ 2 n − 3 containing no 1 -edges and no edges of size at least ( n − 1) , then G and H do not pack if and only if one of G and H has a universal vertex and every vertex of the other is incident to a graph edge. ◮ Our Theorem is sharp even for graphs: for infinitely many n there are n -vertex graphs G n and H n such that | E ( G ) | + | E ( H ) | = 2 n − 2, neither of G n and H n has a universal vertex, and G n and H n do not pack.
Hypergraph Packing ◮ Since each of the graphs in the above two figures has at most 9 vertices, for n ≥ 10 the theorem says that: Let G and H be n-vertex hypergraphs with | E ( G ) | + | E ( H ) | ≤ 2 n − 3 containing no 1 -edges and no edges of size at least ( n − 1) , then G and H do not pack if and only if one of G and H has a universal vertex and every vertex of the other is incident to a graph edge. ◮ Our Theorem is sharp even for graphs: for infinitely many n there are n -vertex graphs G n and H n such that | E ( G ) | + | E ( H ) | = 2 n − 2, neither of G n and H n has a universal vertex, and G n and H n do not pack. ◮ In the same way B. Bollob´ as, S. E. Eldridge’s Theorem yields N. Sauer, J. Spencer’s Theorem, our Main Theorem yields the following extension of N. Sauer, J. Spencer’s Theorem to hypergraphs.
Hypergraph Packing ◮ Corollary: Let G and H be n -vertex hypergraphs with | E ( G ) | + | E ( H ) | < n − 1 + ⌈ n / 2 ⌉ containing no 1-edges and no edges of size at least ( n − 1). Then G and H pack.
Outline of the Proof, Hypergraph Packing ◮ Consider a counterexample ( G , H ) to the Theorem with the least number of vertices n .
Outline of the Proof, Hypergraph Packing ◮ Consider a counterexample ( G , H ) to the Theorem with the least number of vertices n . ◮ This means that | E ( G ) | + | E ( H ) | ≤ 2 n − 3, ( G , H ) is not a bad pair, G and H do not pack, and if one of them has a universal vertex, then the other has a vertex not incident with graph edges.
Outline of the Proof, Hypergraph Packing ◮ Consider a counterexample ( G , H ) to the Theorem with the least number of vertices n . ◮ This means that | E ( G ) | + | E ( H ) | ≤ 2 n − 3, ( G , H ) is not a bad pair, G and H do not pack, and if one of them has a universal vertex, then the other has a vertex not incident with graph edges. ◮ If either G or H is an ordinary graph, then the statement holds by B. Bollob´ as, S. E. Eldridge’s Theorem. So we will assume that each of G and H has at least one hyperedge. (1)
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: (Naroski [2009]) Let G and H be n -vertex hypergraphs with no edge with size less than k and no edge with size greater than n − k . Then there exist n -vertex hypergraphs � G and � H such that
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: (Naroski [2009]) Let G and H be n -vertex hypergraphs with no edge with size less than k and no edge with size greater than n − k . Then there exist n -vertex hypergraphs � G and � H such that ◮ (a) | E ( � G ) | ≤ | E ( G ) | and | E ( � H ) | ≤ | E ( H ) | ,
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: (Naroski [2009]) Let G and H be n -vertex hypergraphs with no edge with size less than k and no edge with size greater than n − k . Then there exist n -vertex hypergraphs � G and � H such that ◮ (a) | E ( � G ) | ≤ | E ( G ) | and | E ( � H ) | ≤ | E ( H ) | , ◮ (b) both � G and � H have no edges of size less than k and no edges of size greater than ⌊ n 2 ⌋ ,
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: (Naroski [2009]) Let G and H be n -vertex hypergraphs with no edge with size less than k and no edge with size greater than n − k . Then there exist n -vertex hypergraphs � G and � H such that ◮ (a) | E ( � G ) | ≤ | E ( G ) | and | E ( � H ) | ≤ | E ( H ) | , ◮ (b) both � G and � H have no edges of size less than k and no edges of size greater than ⌊ n 2 ⌋ , ◮ (c) and, if � G and � H pack, then G and H pack.
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: (Naroski [2009]) Let G and H be n -vertex hypergraphs with no edge with size less than k and no edge with size greater than n − k . Then there exist n -vertex hypergraphs � G and � H such that ◮ (a) | E ( � G ) | ≤ | E ( G ) | and | E ( � H ) | ≤ | E ( H ) | , ◮ (b) both � G and � H have no edges of size less than k and no edges of size greater than ⌊ n 2 ⌋ , ◮ (c) and, if � G and � H pack, then G and H pack. ◮ In view of this lemma, we will assume that G and H have no edges of size greater than n 2 .
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: (Naroski [2009]) Let G and H be n -vertex hypergraphs with no edge with size less than k and no edge with size greater than n − k . Then there exist n -vertex hypergraphs � G and � H such that ◮ (a) | E ( � G ) | ≤ | E ( G ) | and | E ( � H ) | ≤ | E ( H ) | , ◮ (b) both � G and � H have no edges of size less than k and no edges of size greater than ⌊ n 2 ⌋ , ◮ (c) and, if � G and � H pack, then G and H pack. ◮ In view of this lemma, we will assume that G and H have no edges of size greater than n 2 . ◮ We will study properties of the pair ( G , H ) and finally come to a contradiction.
Outline of the Proof; Notations, Hypergraph Packing ◮ Throughout the proof, for i ∈ { 2 , . . . , ⌊ n 2 ⌋} , G i (respectively, H i ) denotes the subgraph of G (respectively, of H ) formed by all of its edges of size i , and d i ( v , G ) (respectively, d i ( v , H )) denotes the degree of vertex v in G i (respectively, in H i ).
Outline of the Proof; Notations, Hypergraph Packing ◮ Throughout the proof, for i ∈ { 2 , . . . , ⌊ n 2 ⌋} , G i (respectively, H i ) denotes the subgraph of G (respectively, of H ) formed by all of its edges of size i , and d i ( v , G ) (respectively, d i ( v , H )) denotes the degree of vertex v in G i (respectively, in H i ). ◮ In particular, G 2 and H 2 are formed by graph edges in G and H , respectively.
Outline of the Proof; Notations, Hypergraph Packing ◮ Throughout the proof, for i ∈ { 2 , . . . , ⌊ n 2 ⌋} , G i (respectively, H i ) denotes the subgraph of G (respectively, of H ) formed by all of its edges of size i , and d i ( v , G ) (respectively, d i ( v , H )) denotes the degree of vertex v in G i (respectively, in H i ). ◮ In particular, G 2 and H 2 are formed by graph edges in G and H , respectively. ◮ Then we let l i := | E ( G i ) | and m i := | E ( H i ) | .
Outline of the Proof; Notations, Hypergraph Packing ◮ Throughout the proof, for i ∈ { 2 , . . . , ⌊ n 2 ⌋} , G i (respectively, H i ) denotes the subgraph of G (respectively, of H ) formed by all of its edges of size i , and d i ( v , G ) (respectively, d i ( v , H )) denotes the degree of vertex v in G i (respectively, in H i ). ◮ In particular, G 2 and H 2 are formed by graph edges in G and H , respectively. ◮ Then we let l i := | E ( G i ) | and m i := | E ( H i ) | . ◮ For brevity, let m := � n i =1 m i , l := � n i =1 l i , m = m − m 1 − m 2 and l = l − l 1 − l 2 .
Outline of the Proof; Notations, Hypergraph Packing ◮ Throughout the proof, for i ∈ { 2 , . . . , ⌊ n 2 ⌋} , G i (respectively, H i ) denotes the subgraph of G (respectively, of H ) formed by all of its edges of size i , and d i ( v , G ) (respectively, d i ( v , H )) denotes the degree of vertex v in G i (respectively, in H i ). ◮ In particular, G 2 and H 2 are formed by graph edges in G and H , respectively. ◮ Then we let l i := | E ( G i ) | and m i := | E ( H i ) | . ◮ For brevity, let m := � n i =1 m i , l := � n i =1 l i , m = m − m 1 − m 2 and l = l − l 1 − l 2 . ◮ In other words, l is the number of hyperedges in G , and m is the number of hyperedges in H .
Outline of the Proof; Notations, Hypergraph Packing ◮ Throughout the proof, for i ∈ { 2 , . . . , ⌊ n 2 ⌋} , G i (respectively, H i ) denotes the subgraph of G (respectively, of H ) formed by all of its edges of size i , and d i ( v , G ) (respectively, d i ( v , H )) denotes the degree of vertex v in G i (respectively, in H i ). ◮ In particular, G 2 and H 2 are formed by graph edges in G and H , respectively. ◮ Then we let l i := | E ( G i ) | and m i := | E ( H i ) | . ◮ For brevity, let m := � n i =1 m i , l := � n i =1 l i , m = m − m 1 − m 2 and l = l − l 1 − l 2 . ◮ In other words, l is the number of hyperedges in G , and m is the number of hyperedges in H . ◮ We always will assume that m ≥ l , and in particular, l ≤ n − 2 . (2)
Outline of the Proof; Notations, Hypergraph Packing ◮ For n -vertex hypergraphs F 1 and F 2 , let x ( F 1 , F 2 ) denote the number of bijections from V ( F 1 ) onto V ( F 2 ) that are not packings.
Outline of the Proof; Notations, Hypergraph Packing ◮ For n -vertex hypergraphs F 1 and F 2 , let x ( F 1 , F 2 ) denote the number of bijections from V ( F 1 ) onto V ( F 2 ) that are not packings. ◮ Since we have chosen G and H that do not pack, x ( G , H ) = n ! . (3)
Outline of the Proof; Notations, Hypergraph Packing ◮ For n -vertex hypergraphs F 1 and F 2 , let x ( F 1 , F 2 ) denote the number of bijections from V ( F 1 ) onto V ( F 2 ) that are not packings. ◮ Since we have chosen G and H that do not pack, x ( G , H ) = n ! . (3) ◮ For edges e ∈ G and f ∈ H , let X ef count the bijections mapping e onto f . ◮ Lemma: ( Naroski [2009]): x ( G , H ) ≤ 2( n − 2)! m 2 l 2 + 3!( n − 3)! ml . (4)
Proof of Naroski’s Lemma, Hypergraph Packing � | x ( G , H ) | = | X ef | e ∈ E ( G ) , f ∈ E ( H )
Proof of Naroski’s Lemma, Hypergraph Packing � | x ( G , H ) | = | X ef | e ∈ E ( G ) , f ∈ E ( H ) � ≤ | X ef | e , f
Proof of Naroski’s Lemma, Hypergraph Packing � | x ( G , H ) | = | X ef | e ∈ E ( G ) , f ∈ E ( H ) � ≤ | X ef | e , f ⌊ n 2 ⌋ � � = | X ef | i =2 e , f : | e | = | f | = i
Proof of Naroski’s Lemma, Hypergraph Packing � | x ( G , H ) | = | X ef | e ∈ E ( G ) , f ∈ E ( H ) � ≤ | X ef | e , f ⌊ n 2 ⌋ � � = | X ef | i =2 e , f : | e | = | f | = i ⌊ n 2 ⌋ � � = i !( n − i )! i =2 e , f : | e | = | f | = i
Proof of Naroski’s Lemma, Hypergraph Packing � | x ( G , H ) | = | X ef | e ∈ E ( G ) , f ∈ E ( H ) � ≤ | X ef | e , f ⌊ n 2 ⌋ � � = | X ef | i =2 e , f : | e | = | f | = i ⌊ n 2 ⌋ � � = i !( n − i )! i =2 e , f : | e | = | f | = i ⌊ n 2 ⌋ � = m i l i i !( n − i )! i =2
Proof of Naroski’s Lemma, Hypergraph Packing ⌊ n 2 ⌋ � | x ( G , H ) |≤ m i l i i !( n − i )! i =2
Proof of Naroski’s Lemma, Hypergraph Packing ⌊ n 2 ⌋ � | x ( G , H ) |≤ m i l i i !( n − i )! i =2 ⌊ n 2 ⌋ � ≤ 2( n − 2)! m 2 l 2 + 3!( n − 3)! m i l i i =3
Proof of Naroski’s Lemma, Hypergraph Packing ⌊ n 2 ⌋ � | x ( G , H ) |≤ m i l i i !( n − i )! i =2 ⌊ n 2 ⌋ � ≤ 2( n − 2)! m 2 l 2 + 3!( n − 3)! m i l i i =3 ⌊ n ⌊ n 2 ⌋ 2 ⌋ � � ≤ 2( n − 2)! m 2 l 2 + 3!( n − 3)! m i l i i =3 i =3
Proof of Naroski’s Lemma, Hypergraph Packing ⌊ n 2 ⌋ � | x ( G , H ) |≤ m i l i i !( n − i )! i =2 ⌊ n 2 ⌋ � ≤ 2( n − 2)! m 2 l 2 + 3!( n − 3)! m i l i i =3 ⌊ n ⌊ n 2 ⌋ 2 ⌋ � � ≤ 2( n − 2)! m 2 l 2 + 3!( n − 3)! m i l i i =3 i =3 = 2( n − 2)! m 2 l 2 + 3!( n − 3)! ml .
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: n ≥ 8.
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: n ≥ 8. ◮ Lemma: m 2 l 2 > ( n − 2) 2 2
Outline of the Proof; Lemmas, Hypergraph Packing ◮ Lemma: n ≥ 8. ◮ Lemma: m 2 l 2 > ( n − 2) 2 2 ◮ Corollary: m 2 > n / 2
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ Suppose that n = 6. According of Naroski’s lemma x ( G , H ) ≤ 2 · 4! m 2 l 2 + (3!) 2 ml .
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ Suppose that n = 6. According of Naroski’s lemma x ( G , H ) ≤ 2 · 4! m 2 l 2 + (3!) 2 ml . ◮ Since m ≥ 1, l ≥ 1 for nonnegative integers m 2 , l 2 and positive integers m , l , the maximum of the expression 2 · 4! m 2 l 2 + (3!) 2 ml under the condition that m 2 + l 2 + m + l ≤ 9 is exactly 6! and is attained only if m 2 = l 2 = 0, l = 4 and m = 5. ◮ So, G and H are 3-uniform hypergraphs with 4 and 5 edges, respectively.
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ Suppose that n = 6. According of Naroski’s lemma x ( G , H ) ≤ 2 · 4! m 2 l 2 + (3!) 2 ml . ◮ Since m ≥ 1, l ≥ 1 for nonnegative integers m 2 , l 2 and positive integers m , l , the maximum of the expression 2 · 4! m 2 l 2 + (3!) 2 ml under the condition that m 2 + l 2 + m + l ≤ 9 is exactly 6! and is attained only if m 2 = l 2 = 0, l = 4 and m = 5. ◮ So, G and H are 3-uniform hypergraphs with 4 and 5 edges, respectively. ◮ Even in this extremal case x ( G , H ) < 6!.
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ Suppose that n = 6. According of Naroski’s lemma x ( G , H ) ≤ 2 · 4! m 2 l 2 + (3!) 2 ml . ◮ Since m ≥ 1, l ≥ 1 for nonnegative integers m 2 , l 2 and positive integers m , l , the maximum of the expression 2 · 4! m 2 l 2 + (3!) 2 ml under the condition that m 2 + l 2 + m + l ≤ 9 is exactly 6! and is attained only if m 2 = l 2 = 0, l = 4 and m = 5. ◮ So, G and H are 3-uniform hypergraphs with 4 and 5 edges, respectively. ◮ Even in this extremal case x ( G , H ) < 6!. ◮ In the proof of Noriski’s Lemma, for every pair of edges e ∈ G and f ∈ H , we considered the cardinality of the set of bijections X ef from V ( G ) onto V ( H ) that map the edge e onto the edge f and estimated Σ := � � f ∈ E ( H ) | X ef | . e ∈ E ( G )
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ Suppose that n = 6. According of Naroski’s lemma x ( G , H ) ≤ 2 · 4! m 2 l 2 + (3!) 2 ml . ◮ Since m ≥ 1, l ≥ 1 for nonnegative integers m 2 , l 2 and positive integers m , l , the maximum of the expression 2 · 4! m 2 l 2 + (3!) 2 ml under the condition that m 2 + l 2 + m + l ≤ 9 is exactly 6! and is attained only if m 2 = l 2 = 0, l = 4 and m = 5. ◮ So, G and H are 3-uniform hypergraphs with 4 and 5 edges, respectively. ◮ Even in this extremal case x ( G , H ) < 6!. ◮ In the proof of Noriski’s Lemma, for every pair of edges e ∈ G and f ∈ H , we considered the cardinality of the set of bijections X ef from V ( G ) onto V ( H ) that map the edge e onto the edge f and estimated Σ := � � f ∈ E ( H ) | X ef | . e ∈ E ( G ) ◮ We will show that some bijection F : V ( G ) → V ( H ) maps at least two edges of G onto two edges of H , thus this bijection counts at least twice in Σ.
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ For this, it is enough to (and we will) find edges e 1 , e 2 ∈ E ( G ) and f 1 , f 2 ∈ E ( H ) such that | e 1 ∩ e 2 | = | f 1 ∩ f 2 | , since in this case we can map e 1 onto f 1 and e 2 onto f 2 .
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ For this, it is enough to (and we will) find edges e 1 , e 2 ∈ E ( G ) and f 1 , f 2 ∈ E ( H ) such that | e 1 ∩ e 2 | = | f 1 ∩ f 2 | , since in this case we can map e 1 onto f 1 and e 2 onto f 2 . ◮ If G has two disjoint edges e and e ′ , then any third edge of G shares one vertex with one of e and e ′ and two vertices with the other. So, we may assume that any two edges in G intersect.
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ For this, it is enough to (and we will) find edges e 1 , e 2 ∈ E ( G ) and f 1 , f 2 ∈ E ( H ) such that | e 1 ∩ e 2 | = | f 1 ∩ f 2 | , since in this case we can map e 1 onto f 1 and e 2 onto f 2 . ◮ If G has two disjoint edges e and e ′ , then any third edge of G shares one vertex with one of e and e ′ and two vertices with the other. So, we may assume that any two edges in G intersect. ◮ Similarly, we may assume that any two edges in H intersect.
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ For this, it is enough to (and we will) find edges e 1 , e 2 ∈ E ( G ) and f 1 , f 2 ∈ E ( H ) such that | e 1 ∩ e 2 | = | f 1 ∩ f 2 | , since in this case we can map e 1 onto f 1 and e 2 onto f 2 . ◮ If G has two disjoint edges e and e ′ , then any third edge of G shares one vertex with one of e and e ′ and two vertices with the other. So, we may assume that any two edges in G intersect. ◮ Similarly, we may assume that any two edges in H intersect. ◮ Now we show that H has a pair of edges with intersection size 1 and a pair of edges with intersection size 2. (5)
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ For this, it is enough to (and we will) find edges e 1 , e 2 ∈ E ( G ) and f 1 , f 2 ∈ E ( H ) such that | e 1 ∩ e 2 | = | f 1 ∩ f 2 | , since in this case we can map e 1 onto f 1 and e 2 onto f 2 . ◮ If G has two disjoint edges e and e ′ , then any third edge of G shares one vertex with one of e and e ′ and two vertices with the other. So, we may assume that any two edges in G intersect. ◮ Similarly, we may assume that any two edges in H intersect. ◮ Now we show that H has a pair of edges with intersection size 1 and a pair of edges with intersection size 2. (5) ◮ If the intersection of each two distinct edges in H contains exactly one vertex, then each vertex belongs to at most two edges, which yields | E ( H ) | ≤ 2 · 6 / 3 = 4, a contradiction to m = 5.
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ Finally, suppose that | f 1 ∩ f 2 | = 2 for all distinct f 1 , f 2 ∈ E ( H ). If two vertices in H , say v 1 and v 2 , are in the intersection of at least three edges, then every other edge also must contain both v 1 and v 2 . Since n = 6 and m = 5, this is impossible.
Proof of Lemma n ≥ 8, Hypergraph Packing ◮ Finally, suppose that | f 1 ∩ f 2 | = 2 for all distinct f 1 , f 2 ∈ E ( H ). If two vertices in H , say v 1 and v 2 , are in the intersection of at least three edges, then every other edge also must contain both v 1 and v 2 . Since n = 6 and m = 5, this is impossible. ◮ Hence we may assume that each pair of vertices is the intersection of at most two edges. Given the edges { v 1 , v 2 , v 3 } and { v 1 , v 2 , v 4 } , every other edge must contain v 3 , v 4 , and one of v 1 or v 2 . Hence each edge of H is contained in { v 1 , v 2 , v 3 , v 4 } . Thus H has at most 4 edges, a contradiction. This proves (5). Hence the lemma holds.
Outline of the Proof; Notations, Hypergraph Packing ◮ Definition: For a hypergraph F without 1-edges and A ⊂ V ( F ), the hypergraph F − A has vertex set V ( F ) − A and E ( F − A ) := { e − A : e ∈ E ( F ) and | e − A | ≥ 2 } , where multiple edges are replaced with a single edge.
Outline of the Proof; Notations, Hypergraph Packing ◮ Definition: For a hypergraph F without 1-edges and A ⊂ V ( F ), the hypergraph F − A has vertex set V ( F ) − A and E ( F − A ) := { e − A : e ∈ E ( F ) and | e − A | ≥ 2 } , where multiple edges are replaced with a single edge. ◮ An edge e of G belongs to a component C of G 2 if strictly more than | e | / 2 vertices of e are in V ( C ).
Outline of the Proof; Notations, Hypergraph Packing ◮ Definition: For a hypergraph F without 1-edges and A ⊂ V ( F ), the hypergraph F − A has vertex set V ( F ) − A and E ( F − A ) := { e − A : e ∈ E ( F ) and | e − A | ≥ 2 } , where multiple edges are replaced with a single edge. ◮ An edge e of G belongs to a component C of G 2 if strictly more than | e | / 2 vertices of e are in V ( C ). ◮ By definition, each e belongs to at most one component of G 2 .
Outline of the Proof; Notations, Hypergraph Packing ◮ Definition: For a hypergraph F without 1-edges and A ⊂ V ( F ), the hypergraph F − A has vertex set V ( F ) − A and E ( F − A ) := { e − A : e ∈ E ( F ) and | e − A | ≥ 2 } , where multiple edges are replaced with a single edge. ◮ An edge e of G belongs to a component C of G 2 if strictly more than | e | / 2 vertices of e are in V ( C ). ◮ By definition, each e belongs to at most one component of G 2 . ◮ A component C of G 2 is clean if no hyperedge belongs to C .
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