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Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

Huajun Zhang

Department of Mathematics, Zhejiang Normal University Zhejiang 321004, P. R. China

December 27, 2013, Shanghai Jiao Tong University

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (EKR Theorem) If A is an intersecting family of k-subsets of [n] = {1, 2, . . . , n}, i.e., A ∩ B = ∅ for any A, B ∈ A, then |A| ≤ n − 1 k − 1

• subject to n ≥ 2k. Equality holds if and only if every subset in A

contains a common element of [n] except for n = 2k.

• P. Erd˝
• s, C. Ko and R. Rado, Intersection theorems for systems of

finite sets, Quart. J. Math. Oxford Ser., 2 (1961), 313-318.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (EKR Theorem for Finite Vector Spaces) If A is an intersecting family of k-dimensional subspaces of an n-dimensional vector space over the q-element field, i.e., dim(A ∩ B) ≥ 1 for any A, B ∈ A, then |A| ≤ n − 1 k − 1

• subject to n ≥ 2k. Equality holds if and only if every subset in A

contains a common nonzero vector except the case n = 2k.

• W. N. Hsieh, Intersection theorems for systems of finite vector

spaces, Discrete Math., 12 (1975), 1-16.

• C. Greene and D. J. Kleitman, Proof techniques in the ordered sets,

in: G.-C. Rota, ed., “Studies in Combinatorics” 1978, 22-79.

• P. Frankl, R. M. Wilson, The Erd˝
• s-Ko-Rado theorem for vector

spaces, JCTA 43 (1986), 228-236.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (EKR Theorem for Permutations) If A is an intersecting family in Sn (the symmetric group on [n]), i.e., for each pair σ, τ ∈ Sn there is an i ∈ [n] with σ(i) = τ(i), then |A| ≤ (n − 1)!. Equality holds if and only if A is a coset of the stabilizer of a point.

• M. Deza and P. Frankl, On the maximum number of permutations

with given maximal or minimal distance, JCTA 22(1977) 352-362.

• P. Cameron and C.Y. Ku, Intersecting families of permutations,

EuJC 24 (2003), 881-890. JW, J. Zhang, An Erd˝

• s-Ko-Rado-type theorem in Coxeter groups,

EuJC 29 (2008), 1112-1115.

• D. Ellis, E. Friedgut, H. Pilpel, Intersecting families of permutations,

Journal of the American Mathematical Society 24 (2011) 649-682.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

A q-signed k-set is a pair (A, f ), where A ⊆ [n] is a k-set and f is a function from A to [q]. A family F of q-signed k-sets is intersecting if for any (A, f ), (B, g) ∈ F there exists x ∈ A ∩ B such that f (x) = g(x). Set Bk

n(q) = {(A, f ) : A ∈

[n]

k

• } and Bn(q) = n

i=0 Bk n(q).

A r-partial permutation of [n] is a pair (A, f ) with A ∈ [n]

r

• and f is an injective map from A to [n].

the set of all r-partial permutations of [n] denoted by Pr,n.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (EKR Theorem for Signed Sets) (Bollob´ as and Leader) Fix a positive integer k ≤ n, and let F be an intersecting family of q-signed k-sets on [n], where q ≥ 2. Then |F| ≤ n−1

k−1

• qk−1. Unless q = 2 and k = n, equality holds if and
• nly if F consists of all q-signed k-sets (A, f ) such that x0 ∈ A

and f (x0) = ε0 for some fixed x0 ∈ [n], ε0 ∈ [q].

Bollobas B. Bollob´ as and I. Leader, An Erd¨

• s-Ko-Rado theorem for

signed sets, Comput. Math. Applic. 34 (11) (1997) 9-13. Y.S. Li, J. Wang, Erd˝

• s-Ko-Rado-Type Theorems for Colored Sets,
• Electron. J. Combin. 14 (1) (2007).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (EKR Theorem for Partial Permutation) Fix a positive integer r < n, and let F be an intersecting family of Pr,n. Then |F| ≤ n−1

r−1

(n − 1)! (n − r)!. Equality holds if and only if F consists of all r-partial permutations (A, f ) such that i ∈ A and f (i) = j for some fixed i, j ∈ [n].

• C. Y. Ku and I. Leader, An Erd¨
• s-Ko-Rado theorem for partial

permutations, Disc. Math. 306 (2006) 74-86. Y.S. Li, J. Wang, Erd˝

• s-Ko-Rado-Type Theorems for Colored Sets,
• Electron. J. Combin. 14 (1) (2007).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (Hilton, 1977) Let A1, A2, . . . , Am be cross-intersecting families of k subsets of [n] with A1 = ∅, i.e., for any Ai ∈ Ai and Aj ∈ Aj, i = j, Ai ∩ Aj = ∅. If k ≤ n/2, then

m

• i=1

|Ai| ≤

• n

k

• ,

if m ≤ n/k; m n−1

k−1

• ,

if m ≥ n/k. Unless m = 2 = n/k, the bound is attained if and only if one of the following holds: (i) m ≤ n/k and A1 = [n]

k

• , and A2 = · · · = Am = ∅;

(ii) m ≥ n/k and |A1| = |A2| = . . . = |Am| = n−1

k−1

• .

A.J.W. Hilton, An intersection theorem for a collection of families of subsets of a finite set, J. London Math. Soc. 2 (1977) 369-384.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Results The Hilton Theorem was generalized to partial permutation, signed sets and labled sets.

• P. Borg, Cross-intersecting families of permutations, J. Combin.

Theory Ser. A, 117 (2010) 483-487.

• P. Borg, Intersecting and cross-independent families of labeled sets,
• Electron. J. Combin. 15 (2008) N9.
• P. Borg and I. Leader, Multiple cross-intersecting families of signed

sets, J. Combin. Theory Ser. A, 117 (2010) 583-588.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Results The Hilton Theorem was generalized to general case.

• J. Wang, H.J. Zhang, Cross-intersecting families and primitivity of

symmetric systems, J. Combin. Theory Ser. A 118 (2011) 455-462.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (Hilton and Milner 1967) Let n and a be two positive integers with n ≥ 2a. If A, B ⊆ [n]

a

• with A ∩ B = ∅ for all A ∈ A and B ∈ B, then

|A| + |B| ≤ n a

• −

n − a a

• + 1.

A.J.W. Hilton and E.C. Milner, Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford 18 (1967) 369-384.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem ( Frankl and Tohushige) Let n, a and b be three positive integers with n ≥ a + b and a ≤ b. If A ⊆ [n]

a

• and B ⊆

[n]

b

• with A ∩ B = ∅ for all A ∈ A and

B ∈ B, then |A| + |B| ≤ n b

• −

n − a b

• + 1.
• P. Frankl and N. Tohushige, Some best possible inequalities

concerning cross-intersecting families, J. Combin. Theory Ser. A 61 (1992) 87-97.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Results The Hilton-Milner Theorem was generalized to the general cases.

• J. Wang, H.J. Zhang, Nontrivial independent sets of bipartite

graphs and cross-intersecting families, J. Combin. Theory Ser. A, 120 (2013) 129-141.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem If A ⊆ [n]

k

• and B ⊆

[n]

ℓ

• are cross-intersecting with k, ℓ ≤ n/2,

then |A||B| ≤ n − 1 k − 1 n − 1 ℓ − 1

• .

Moreover, the equality holds if and only if A = {A ∈ [n]

k

• : i ∈ A}

and B = {B ∈ [n]

ℓ

• : i ∈ B} for some i ∈ [n], unless n = 2k = 2ℓ.
• L. Pyber, A new generalization of the Erd˝
• s-Rado-Ko theorem, J.
• Combin. Theory Ser. A, 43 (1986) 85-90.
• M. Matsumoto, N. Tokushige, The exact bound in the

Erd˝

• s-Rado-Ko theorem for cross-intersecting families, J. Combin.

Theory Ser. A, 52 (1989) 90-97.

• C. Bey, On cross-intersecting families of sets, Graphs Combin., 21

(2005) 161-168.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (Tokushige) Let p be a real with 0 < p < 0.114, and let t be an integer with 1 ≤ t ≤ 1/(2p). For fixed p and t there exist positive constants ε, n1 such that for all integers n, k with n > n1 and |k n − p| < ε,the following is true: if two families A1 ⊂ [n]

k

• and A2 ⊂

[n]

k

• are

cross t-intersecting, then |A1||A2| ≤ n − t k − t 2 with equality holding iff A1 = A2 = {F ∈ [n]

k

• : [t] ⊂ F} (up to

isomorphism).

• N. Tokushige, On cross t-intersecting families of sets, J. Combin.

Theory Ser. A, 117 (2010)1167-1177.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem (Ellis, Friedgut, Pilpel) For any positive integer k and any n sufficiently large depending on k, if I, J ⊂ Sn are k-cross-intersecting, then |I||J| ≤ ((n − k)!)2. Equality holds if and only if I = J and I is a k-coset of Sn.

• D. Ellis, E. Friedgut, H. Pilpel, Intersecting families of permutations,

Journal of the American Mathematical Society 24 (2011) 649-682.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 1. Erd˝
• s-Ko-Rado Theorem

Theorem Let n and p be two positive integer with p ≥ 4. If A and B are cross-intersecting families in Lp =

• {(1, ℓ1), (2, ℓ2), . . . , (n, ℓn)} : ℓi ∈ [p], i = 1, 2 . . . , n
• , then

|A||B| ≤ p2n−2, and equality holds if and only if A = B =

• {(1, ℓ1), (2, ℓ2), . . . , (n, ℓn)} : ℓi = j
• for some i ∈ [n]

and j ∈ [p].

H.J. Zhang, Cross-intersecting families in labeled sets, Electron. J.

• Combin. 20 (2013).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

Let G be a simple graph with vertex set V (G) and edge set E(G). S ⊆ V (G). S is an independent set of G if no two elements of S are adjacent in G.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

Let G be a simple graph with vertex set V (G) and edge set E(G). S ⊆ V (G). S is an independent set of G if no two elements of S are adjacent in G. S is a k-independent set of G if S can be expressed as a union

• f k independent sets of G.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

Let G be a simple graph with vertex set V (G) and edge set E(G). S ⊆ V (G). S is an independent set of G if no two elements of S are adjacent in G. S is a k-independent set of G if S can be expressed as a union

• f k independent sets of G.

k-independence number αk(G): the maximum size of k-independent sets of G,

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

The Kneser graph K(n, k)(2k ≤ n): vertex set [n]

k

• and A ∼ B iff

A ∩ B = ∅. Erd˝

• s-Ko-Rado: α1(K(n, k)) =

n−1

k−1

• .

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

The Kneser graph K(n, k)(2k ≤ n): vertex set [n]

k

• and A ∼ B iff

A ∩ B = ∅. Erd˝

• s-Ko-Rado: α1(K(n, k)) =

n−1

k−1

• .

Given a graph G, what is α1(G)?

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

Given a graph G, what is αk(G)?

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

Note that A = {A ∈ [n]

k

• : A ∩ {1, 2} = ∅} is a 2-independent set
• f K(n, k).

Someone conjectured that α2(K(n, k)) = n−1

k−1

• +

n−2

k−1

• for

sufficient larger n.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

Note that A = {A ∈ [n]

k

• : A ∩ {1, 2} = ∅} is a 2-independent set
• f K(n, k).

Someone conjectured that α2(K(n, k)) = n−1

k−1

• +

n−2

k−1

• for

sufficient larger n. C.Godsil et al: α2(K(9, 4)) = 95 or 96

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

Example A =

• A ∈

[9] 4

• : 1 ∈ A and A ∩ {2, 3, 4} = ∅
• B =
• B ∈

[9] 4

• : 1 ∈ A and {2, 3, 4} ⊂ B
• C =
• C ∈

[9] 4

• : |C ∩ {5, 6, 7, 8, 9}| ≥ 3
• Clearly, A ∪ B and C are two disjoint independent sets of K(9, 4),

and so A ∪ B ∪ C is a 2-independent set. |A| + |B| + |C| = 8 3

• −

5 3

• +

5 1

• +

5 4

• +

5 3 4 1

• = 96.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

Example A =

• A ∈

[9] 4

• : 1 ∈ A and A ∩ {2, 3} = ∅
• B =
• B ∈

[9] 4

• : 1 ∈ A and {2, 3} ⊂ B
• C =
• C ∈

[9] 4

• : |C ∩ {5, 6, 7, 8, 9}| ≥ 3
• A ∪ B ∪ C is also a 2-independent set.

|A| + |B| + |C| = 96.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

For n = 2k + 1, we conjecture

α2(K(n, k)) = n − 1 k − 1

• +

n − 2 k − 1

• +

          

• i≥⌈k/2⌉+1

k i k k − i

• ,

if k is odd;

• i≥⌈k/2⌉+1

k − 1 i k + 1 k − i

• ,

if k is even.

α2(K(9, 4)) = 96.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 2. 2-independent sets in Kneser graphs

For n > 2k + 1, we conjecture α2(K(n, k)) = n − 1 k − 1

• +

n − 2 k − 1

• .

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Theorem (Tur´ an 1941) Let G(V , E) be a graph on n vertices without k-clique, then |E| ≤ (k − 2)n2 2(k − 1) .

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

A hypergraph H is a pair H = (X, E) where X is a set of elements, called vertices, and E is a set of non-empty subsets

• f X called edges.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

A hypergraph H is a pair H = (X, E) where X is a set of elements, called vertices, and E is a set of non-empty subsets

• f X called edges.

H is said to be k-uniform if |E| = k for all E ∈ E.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

A hypergraph H is a pair H = (X, E) where X is a set of elements, called vertices, and E is a set of non-empty subsets

• f X called edges.

H is said to be k-uniform if |E| = k for all E ∈ E. H is said to be complete k-uniform if E = X

k

• .

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

A hypergraph H is a pair H = (X, E) where X is a set of elements, called vertices, and E is a set of non-empty subsets

• f X called edges.

H is said to be k-uniform if |E| = k for all E ∈ E. H is said to be complete k-uniform if E = X

k

• .

A family {E1, E2, . . . , Es} ⊂ E is called a matching if Ei’s are pairwise disjoint.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

A hypergraph H is a pair H = (X, E) where X is a set of elements, called vertices, and E is a set of non-empty subsets

• f X called edges.

H is said to be k-uniform if |E| = k for all E ∈ E. H is said to be complete k-uniform if E = X

k

• .

A family {E1, E2, . . . , Es} ⊂ E is called a matching if Ei’s are pairwise disjoint. For F ⊆ E, the matching number ν(F) is the size of the maximum matching contained in F.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Erd˝

• s posed the following conjecture.

Conjecture If F ⊂ [n]

k

• , ν(F) = s and n ≥ (s + 1)k, then

|F| ≤ max (sk + k − 1 k

• ,

n k

• −

n − s k

• .

(1) The case s = 1 is the classical Erd˝

• s-Ko-Rado Theorem.
• P. Erd˝
• s, A problem on independent r-tuples, Ann. Univ. Sci.
• Budapest. Eotvos Sect. Math., 8 (1965) 93–95.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Erd˝

• s proved that the conjecture holds if n > n0(k, s).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Erd˝

• s proved that the conjecture holds if n > n0(k, s).

Bollob´ as, Daykin and Erd˝

• s (1976) proved that

n0(k, s) = 2sk3.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Erd˝

• s proved that the conjecture holds if n > n0(k, s).

Bollob´ as, Daykin and Erd˝

• s (1976) proved that

n0(k, s) = 2sk3. F¨ uredi and Frankl (unpublished) proved that n0(k, s) = 100k2.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Erd˝

• s proved that the conjecture holds if n > n0(k, s).

Bollob´ as, Daykin and Erd˝

• s (1976) proved that

n0(k, s) = 2sk3. F¨ uredi and Frankl (unpublished) proved that n0(k, s) = 100k2. Huang, Loh and Sudakov (2012) proved that n0(k, s) = 3k2.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Erd˝

• s proved that the conjecture holds if n > n0(k, s).

Bollob´ as, Daykin and Erd˝

• s (1976) proved that

n0(k, s) = 2sk3. F¨ uredi and Frankl (unpublished) proved that n0(k, s) = 100k2. Huang, Loh and Sudakov (2012) proved that n0(k, s) = 3k2. Frankl(2013) proved that n0(k, s) = 2(s + 1)k − s.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Erd˝

• s proved that the conjecture holds if n > n0(k, s).

Bollob´ as, Daykin and Erd˝

• s (1976) proved that

n0(k, s) = 2sk3. F¨ uredi and Frankl (unpublished) proved that n0(k, s) = 100k2. Huang, Loh and Sudakov (2012) proved that n0(k, s) = 3k2. Frankl(2013) proved that n0(k, s) = 2(s + 1)k − s. The cases k = 2 and k = 3 were settled by Erd˝

• s and Gallai

(1959) and Frankl (recently),resp.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

A linear path of length ℓ is a family of sets {F1, F2, . . . , Fℓ} such that |Fi ∩ Fi+1| = 1 for each i and Fi ∩ Fj = ∅ whenever |i − j| > 1. Let P(k)

ℓ

denote the k-uniform linear path of length of ℓ. Theorem (F¨ uredi, Jiang and Seiver) Let k, t be positive integers, where k ≥ 3. For sufficiently larger n, we have exk(n; P(k)

2t+1) =

n − 1 k − 1

• +

n − 2 k − 1

• + . . . +

n − t k − 1

• .

The only extremal family consists of all the k-sets in [n] that meet some fixed set S of t vertices.

• Z. F¨

uredi, T. Jiang and R. Seiver, Exact solution of the hypergraph turan problem for k-uniform linear paths, Combinatorica, to appear.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Theorem (F¨ uredi, Jiang and Seiver) Let k, t be positive integers, where k ≥ 3. For sufficiently larger n, we have exk(n; P(k)

2t+2) =

n − 1 k − 1

• +

n − 2 k − 1

• +. . .+

n − t k − 1

• +

n − t − 2 k − 2

• .

The only extremal family consists of all the k-sets in [n] that meet some fixed set S of t vertices plus all the k-sets in [n] \ S that contain some two fixed elements.

• Z. F¨

uredi, T. Jiang and R. Seiver, Exact solution of the hypergraph turan problem for k-uniform linear paths, Combinatorica, to appear.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

A k-uniform minimal cycle of length ℓ is a cyclic list of k-sets A1, A2, . . . , Aℓ such that consecutive sets intersect in at least

• ne element and nonconsecutive sets are disjoint. The set of

all k-uniform minimal cycles of length ℓ denoted by C(k)

ℓ

.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

A k-uniform minimal cycle of length ℓ is a cyclic list of k-sets A1, A2, . . . , Aℓ such that consecutive sets intersect in at least

• ne element and nonconsecutive sets are disjoint. The set of

all k-uniform minimal cycles of length ℓ denoted by C(k)

ℓ

. A k-uniform linear cycle of length ℓ, denoted by C(k)

ℓ

is a cyclic list of k-sets A1, A2, . . . , Aℓ such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Theorem (F¨ uredi and Jiang) Let t b e a positive integer, k ≥ 4. For sufficiently larger n, we have exk(n, C(k)

2t+1) =

n k

• −

n − t k

• and

exk(n, C(k)

2t+2) =

n k

• −

n − t k

• + 1.

The only extremal family consists of all the k-sets in [n] that meet some fixed k-set S. For C(k)

2t+2, the only extremal family consists of

all the k-sets in [n] that meet some fixed k-set S plus one additional k-set outside S.

• Z. F¨

uredi and T. Jiang, Hypergraph turan numbers of linear cycles, arXiv:1302.2387.[math.CO]2013.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Theorem (F¨ uredi and Jiang) Let t b e a positive integer, k ≥ 5. For sufficiently larger n, we have exk(n, C(k)

2t+1) =

n k

• −

n − t k

• and

exk(n, C(k)

2t+2) =

n k

• −

n − t k

• +

n − t − 2 k − 2

• .

For C(k)

2t+1, the only extremal family consists of all the k-sets in [n]

that meet some fixed k-set S. For C(k)

2t+2, the only extremal family

consists of all the k-sets in [n] that meet some fixed k-set S plus all the k-sets in [n] \ S that contain some two two fixed elements.

• Z. F¨

uredi and T. Jiang, Hypergraph turan numbers of linear cycles, arXiv:1302.2387.[math.CO]2013.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

S = {C1 ∪ . . . ∪ Cr : Ci ∈ C(k)

ℓi

for i ∈ [r]} Theorem (Gu, Li and Shi) Let integers k ≥ 4, r ≥ 1, ℓ1, . . . , ℓr ≥ 3, t =

r

• i=1

⌊ℓi + 1 2 ⌋ − 1, and I = 1 if all the ℓ1, . . . , ℓr are even, I = 0 otherwise. For sufficiently large n, exk(n; S(ℓ1, . . . , ℓr)) = n k

• −

n − t k

• + I.

Ran Gu, Xueliang Li and Yongtang Shi, Hypergraph turan numbers of vertex disjoint cycles, arXive: 1305.5372v1 [math.CO]2013.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Theorem (Gu, Li and Shi) Let integers k ≥ 5, r ≥ 1, ℓ1, . . . , ℓr ≥ 3, t =

r

• i=1

⌊ℓi + 1 2 ⌋ − 1, and J = n−t−2

k

• if all the ℓ1, . . . , ℓr are even, J = 0 otherwise. For

sufficiently large n, exk(n; C(k)

ℓ1 , . . . , C(k) ℓr ) =

n k

• −

n − t k

• + J.

Ran Gu, Xueliang Li and Yongtang Shi, Hypergraph turan numbers of vertex disjoint cycles, arXive: 1305.5372v1 [math.CO]2013.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Bk

n(q) = {(A, f ) : A ∈

[n]

k

• }.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Bk

n(q) = {(A, f ) : A ∈

[n]

k

• }.

Theorem For positive integers q, n and k with k ≤ n, if F ⊂ Bk

n(q) with

ν(F) = s where s < q, then |F| ≤ sqk−1 n − 1 k − 1

• ,

and equality holds if and only if F is isomorphic to F1 = {(A, f ) ∈ Bk

n(q) : 1 ∈ A, f (1) ∈ [s]}.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Let V be a finite set and let p be an ideal of 2V , that is, p consists of subsets of V such that B ∈ p if B ⊆ A for some A ∈ p.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Let V be a finite set and let p be an ideal of 2V , that is, p consists of subsets of V such that B ∈ p if B ⊆ A for some A ∈ p. Define αp(V ) := max{|A| : A ∈ p} and write M(V ) = {A ∈ p : |A| = αp(V )}

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Let V be a finite set and let p be an ideal of 2V , that is, p consists of subsets of V such that B ∈ p if B ⊆ A for some A ∈ p. Define αp(V ) := max{|A| : A ∈ p} and write M(V ) = {A ∈ p : |A| = αp(V )} For H ⊂ V , write αp(H) := max{|A| : A ∈ p and A ⊆ H}.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Let V be a finite set and let p be an ideal of 2V , that is, p consists of subsets of V such that B ∈ p if B ⊆ A for some A ∈ p. Define αp(V ) := max{|A| : A ∈ p} and write M(V ) = {A ∈ p : |A| = αp(V )} For H ⊂ V , write αp(H) := max{|A| : A ∈ p and A ⊆ H}. For positive integer s, set p = {F ⊆ Bk

n(q) : ν(F) ≤ s}.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

To complete the proof, we only need to determine αp(Bk

n (q)) and

M(Bk

n (q)).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Lemma Let V be a finite set and p an idea on 2V . Suppose that there is a transitive permutation group Γ on V that keeps the ideal p, i.e., σ(A) ∈ p for all A ∈ p and σ ∈ Γ. Then, for each H ⊆ V , αp(V ) |V | ≤ αp(H) |H| , (2) and equality holds if and only if |H ∩ S| = αp(H) for each S ∈ M(V ).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Bk

n(q) = {(A, f ) : A ∈

[n]

k

• and f is an injection from A to [q]}

1 2 3 . . . n 1 2 3 . . . n . . . 1 2 3 . . . n 1 1 1 . . . 1 2 2 2 . . . 2 . . . q q q . . . q

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Bk

n(q) = {(A, f ) : A ∈

[n]

k

• and f is an injection from A to [q]}

1 2 3 . . . n 1 2 3 . . . n . . . 1 2 3 . . . n 1 1 1 . . . 1 2 2 2 . . . 2 . . . q q q . . . q (A, f ) is said to be contained in the above cycle if it consists

• f k consecutive elements.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Bk

n(q) = {(A, f ) : A ∈

[n]

k

• and f is an injection from A to [q]}

1 2 3 . . . n 1 2 3 . . . n . . . 1 2 3 . . . n 1 1 1 . . . 1 2 2 2 . . . 2 . . . q q q . . . q (A, f ) is said to be contained in the above cycle if it consists

• f k consecutive elements.

Let H be the set of (A, f ) which contained in the above cycle. Clearly |H| = nq.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Bk

n(q) = {(A, f ) : A ∈

[n]

k

• and f is an injection from A to [q]}

1 2 3 . . . n 1 2 3 . . . n . . . 1 2 3 . . . n 1 1 1 . . . 1 2 2 2 . . . 2 . . . q q q . . . q (A, f ) is said to be contained in the above cycle if it consists

• f k consecutive elements.

Let H be the set of (A, f ) which contained in the above cycle. Clearly |H| = nq. Determine αp(H).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Graph G[H]: V (G[H]) = H, two vertices (A, f ) and (B, g) are adjacent iff they are not intersecting.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Graph G[H]: V (G[H]) = H, two vertices (A, f ) and (B, g) are adjacent iff they are not intersecting. G[H] is isomorphism to the well-known circular graph Knq:k.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Graph G[H]: V (G[H]) = H, two vertices (A, f ) and (B, g) are adjacent iff they are not intersecting. G[H] is isomorphism to the well-known circular graph Knq:k. Clique number ω(G): the maximum number of pairwise adjacent vertices in G. For S ⊆ V (G), ω(G[S]) written as ω(S).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Graph G[H]: V (G[H]) = H, two vertices (A, f ) and (B, g) are adjacent iff they are not intersecting. G[H] is isomorphism to the well-known circular graph Knq:k. Clique number ω(G): the maximum number of pairwise adjacent vertices in G. For S ⊆ V (G), ω(G[S]) written as ω(S). F of H with ν(F) = s iff ω(F) = s.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Graph G[H]: V (G[H]) = H, two vertices (A, f ) and (B, g) are adjacent iff they are not intersecting. G[H] is isomorphism to the well-known circular graph Knq:k. Clique number ω(G): the maximum number of pairwise adjacent vertices in G. For S ⊆ V (G), ω(G[S]) written as ω(S). F of H with ν(F) = s iff ω(F) = s. To determine αp(H) is equivalent to determine ω(G[H]).

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Kn:k: vertex set [n], i ∼ j iff k ≤ |i − j| ≤ n − k.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Kn:k: vertex set [n], i ∼ j iff k ≤ |i − j| ≤ n − k. The maximum independent sets in Kn:k are the sets of k consecutive elements, viewed cyclically.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Kn:k: vertex set [n], i ∼ j iff k ≤ |i − j| ≤ n − k. The maximum independent sets in Kn:k are the sets of k consecutive elements, viewed cyclically. Lemma Let n, s and k be three integers with n ≥ k(s + 1). For any vertex subset F of Kn:k with ω(F) = s, then |F| ≤ ks. Moreover, if n > k(s + 1), the equality holds if and only if F is the union of s maximum independent sets of Kn:k.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

ω(G[H]) = ks Let F be a family of Bk

n(q) with ν(F) = s. Then by the former

lemmas we have |F| |Bk

n(q)| ≤ sk

nq . Therefore, |F| ≤ sk nq qk n k

• = sqk−1

n − 1 k − 1

• .

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

For n positive numbers p1, p2, . . . , pn with p1 ≤ p2 ≤ · · · ≤ pn, let Lp be the labeled n-sets given by Ln,p = {(i1, i2, . . . , in) : ij ∈ [pj] for j ∈ [n]}. Theorem If F is a family of Ln,p with ν(F) = s ≤ p1, then |F| ≤ sp2p3 · · · pn, and equality holds if and only if F = {(i1, i2, . . . , in) ∈ Ln,p : ij ∈ S} for one s-subset S of [p1] and

• ne j of [n] with pj = p1.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Theorem Let n and s be two positive integers with s ≤ n. If F is a family of Sn with ν(F) = s, then |F| ≤ s(n − 1)!.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

• 3. Matching Numbers

Theorem Let n and s be two positive integers with s ≤ n. If F is a family of Sn with ν(F) = s, then |F| ≤ s(n − 1)!. Theorem Let n and s be two positive integers with s < n. If F is a family of Pr,n with ν(F) = s, then |F| ≤ s n−1

r−1

(n − 1)! (n − r)!.

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem

Many Thanks!

Huajun Zhang Some Problems and Generalizations on Erd˝

• s-Ko-Rado Theorem