Some Problems and Generalizations on Erdos-Ko-Rado …1. Erd}os-Ko-Rado Theorem Theorem (EKR Theorem...

Some Problems and Generalizations onErdos-Ko-Rado Theorem

Huajun Zhang

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

December 27, 2013, Shanghai Jiao Tong University

Huajun Zhang Some Problems and Generalizations on Erdos-Ko-Rado Theorem

1. Erdos-Ko-Rado Theorem

Theorem (EKR Theorem)

If A is an intersecting family of k-subsets of [n] = {1, 2, . . . , n},i.e., A ∩ B 6= ∅ for any A,B ∈ A, then

|A| ≤(

n − 1

k − 1

)subject to n ≥ 2k. Equality holds if and only if every subset in Acontains a common element of [n] except for n = 2k.

P. Erdos, C. Ko and R. Rado, Intersection theorems for systems offinite sets, Quart. J. Math. Oxford Ser., 2 (1961), 313-318.



Theorem (EKR Theorem for Finite Vector Spaces)

If A is an intersecting family of k-dimensional subspaces of ann-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 Acontains a common nonzero vector except the case n = 2k.

W. N. Hsieh, Intersection theorems for systems of finite vectorspaces, 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 Erdos-Ko-Rado theorem for vectorspaces, JCTA 43 (1986), 228-236.



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 permutationswith 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 Erdos-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.



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

Set Bkn(q) = {(A, f ) : A ∈([n]k

)} and Bn(q) =

⋃ni=0 Bkn(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.



Theorem (EKR Theorem for Signed Sets)

(Bollobas and Leader) Fix a positive integer k ≤ n, and let F bean intersecting family of q-signed k-sets on [n], where q ≥ 2. Then|F| ≤

(n−1k−1

)qk−1. Unless q = 2 and k = n, equality holds if and

only if F consists of all q-signed k-sets (A, f ) such that x0 ∈ Aand f (x0) = ε0 for some fixed x0 ∈ [n], ε0 ∈ [q].

Bollobas B. Bollobas and I. Leader, An Erdos-Ko-Rado theorem forsigned sets, Comput. Math. Applic. 34 (11) (1997) 9-13.

Y.S. Li, J. Wang, Erdos-Ko-Rado-Type Theorems for Colored Sets,Electron. J. Combin. 14 (1) (2007).



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−1r−1

)(n − 1)!

(n − r)!. Equality holds if and only if F

consists of all r -partial permutations (A, f ) such that i ∈ A andf (i) = j for some fixed i , j ∈ [n].

C. Y. Ku and I. Leader, An Erdos-Ko-Rado theorem for partialpermutations, Disc. Math. 306 (2006) 74-86.

Y.S. Li, J. Wang, Erdos-Ko-Rado-Type Theorems for Colored Sets,Electron. J. Combin. 14 (1) (2007).



Theorem (Hilton, 1977)

Let A1,A2, . . . ,Am be cross-intersecting families of k subsets of[n] with A1 6= ∅, i.e., for any Ai ∈ Ai and Aj ∈ Aj , i 6= j ,Ai ∩ Aj 6= ∅. If k ≤ n/2, then

m∑i=1

|Ai | ≤{ (n

k

), if m ≤ n/k;

m(n−1k−1

), if m ≥ n/k.

Unless m = 2 = n/k, the bound is attained if and only if one ofthe following holds:

(i) m ≤ n/k and A1 =([n]k

), and A2 = · · · = Am = ∅;

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

).

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



Results

The Hilton Theorem was generalized to partial permutation, signedsets 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 signedsets, J. Combin. Theory Ser. A, 117 (2010) 583-588.



Results

The Hilton Theorem was generalized to general case.

J. Wang, H.J. Zhang, Cross-intersecting families and primitivity ofsymmetric systems, J. Combin. Theory Ser. A 118 (2011) 455-462.



Theorem (Hilton and Milner 1967)

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

)with A ∩ B 6= ∅ 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 forsystems of finite sets, Quart. J. Math. Oxford 18 (1967) 369-384.



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 6= ∅ for all A ∈ A and

B ∈ B, then

|A|+ |B| ≤(

n

b

)−(

n − a

b

)+ 1.

P. Frankl and N. Tohushige, Some best possible inequalitiesconcerning cross-intersecting families, J. Combin. Theory Ser. A 61(1992) 87-97.



Results

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

J. Wang, H.J. Zhang, Nontrivial independent sets of bipartitegraphs and cross-intersecting families, J. Combin. Theory Ser. A,120 (2013) 129-141.



Theorem

If A ⊆([n]k

)and B ⊆

([n]`

)are cross-intersecting with k , ` ≤ n/2,

then

|A||B| ≤(

n − 1

k − 1

)(n − 1

`− 1

).

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

): i ∈ A}

and B = {B ∈([n]

`

): i ∈ B} for some i ∈ [n], unless n = 2k = 2`.

L. Pyber, A new generalization of the Erdos-Rado-Ko theorem, J.Combin. Theory Ser. A, 43 (1986) 85-90.

M. Matsumoto, N. Tokushige, The exact bound in theErdos-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.



Theorem (Tokushige)

Let p be a real with 0 < p < 0.114, and let t be an integer with1 ≤ 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 |kn− 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.



Theorem (Ellis, Friedgut, Pilpel)

For any positive integer k and any n sufficiently large depending onk, 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.



Theorem

Let n and p be two positive integer with p ≥ 4. If A and B arecross-intersecting families inLp =

{{(1, `1), (2, `2), . . . , (n, `n)} : ì ∈ [p], i = 1, 2 . . . , n

}, then

|A||B| ≤ p2n−2,

and equality holds if and only ifA = B =

{{(1, `1), (2, `2), . . . , (n, `n)} : ì = j

}for some i ∈ [n]

and j ∈ [p].

H.J. Zhang, Cross-intersecting families in labeled sets, Electron. J.Combin. 20 (2013).


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 areadjacent in G .

S is a k-independent set of G if S can be expressed as a unionof k independent sets of G .

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















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

)and A ∼ B iff

A ∩ B = ∅.Erdos-Ko-Rado: α1(K (n, k)) =

(n−1k−1

).

Given a graph G , what is α1(G )?



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

)and A ∼ B iff

A ∩ B = ∅.Erdos-Ko-Rado: α1(K (n, k)) =

(n−1k−1

).

Given a graph G , what is α1(G )?



Given a graph G , what is αk(G )?



Note that A = {A ∈([n]k

): A ∩ {1, 2} 6= ∅} is a 2-independent set

of K (n, k).

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

)+(n−2k−1

)for

sufficient larger n.

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



Note that A = {A ∈([n]k

): A ∩ {1, 2} 6= ∅} is a 2-independent set

of K (n, k).

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

)+(n−2k−1

)for

sufficient larger n.

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



Example

A =

{A ∈

([9]

4

): 1 ∈ A and A ∩ {2, 3, 4} 6= ∅

}B =

{B ∈

([9]

4

): 1 6∈ 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.



Example

A =

{A ∈

([9]

4

): 1 ∈ A and A ∩ {2, 3} 6= ∅

}B =

{B ∈

([9]

4

): 1 6∈ 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.



For n = 2k + 1, we conjecture

α2(K (n, k)) =

(n − 1

k − 1

)+

(n − 2

k − 1

)

+

∑

i≥dk/2e+1

(k

i

)(k

k − i

), if k is odd;

∑i≥dk/2e+1

(k − 1

i

)(k + 1

k − i

), if k is even.

α2(K (9, 4)) = 96.



For n > 2k + 1, we conjecture

α2(K (n, k)) =

(n − 1

k − 1

)+

(n − 2

k − 1

).


3. Matching Numbers

Theorem (Turan 1941)

Let G (V ,E ) be a graph on n vertices without k-clique, then

|E | ≤ (k − 2)n2

2(k − 1).


3. Matching Numbers

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

).

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

For F ⊆ E , the matching number ν(F) is the size of themaximum matching contained in F .


3. Matching Numbers




).




3. Matching Numbers




).




3. Matching Numbers




).




3. Matching Numbers




).




3. Matching Numbers

Erdos 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 Erdos-Ko-Rado Theorem.

P. Erdos, A problem on independent r-tuples, Ann. Univ. Sci.Budapest. Eotvos Sect. Math., 8 (1965) 93–95.


3. Matching Numbers

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

Bollobas, Daykin and Erdos (1976) proved thatn0(k , s) = 2sk3.

Furedi 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 Erdos and Gallai(1959) and Frankl (recently),resp.


3. Matching Numbers








3. Matching Numbers








3. Matching Numbers








3. Matching Numbers








3. Matching Numbers








3. Matching Numbers

A linear path of length ` is a family of sets {F1,F2, . . . ,F`}such that |Fi ∩ Fi+1| = 1 for each i and Fi ∩ Fj = ∅ whenever

|i − j | > 1. Let P(k)` denote the k-uniform linear path of

length of `.

Theorem (Furedi, 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 meetsome fixed set S of t vertices.

Z. Furedi, T. Jiang and R. Seiver, Exact solution of thehypergraph turan problem for k-uniform linear paths,Combinatorica, to appear.


3. Matching Numbers

Theorem (Furedi, 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 meetsome fixed set S of t vertices plus all the k-sets in [n] \ S thatcontain some two fixed elements.

Z. Furedi, T. Jiang and R. Seiver, Exact solution of thehypergraph turan problem for k-uniform linear paths,Combinatorica, to appear.


3. Matching Numbers

A k-uniform minimal cycle of length ` is a cyclic list of k-setsA1,A2, . . . ,A` such that consecutive sets intersect in at leastone element and nonconsecutive sets are disjoint. The set of

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

A k-uniform linear cycle of length `, denoted by C(k)` is a

cyclic list of k-sets A1,A2, . . . ,A` such that consecutive setsintersect in exactly one element and nonconsecutive sets aredisjoint.


3. Matching Numbers

A k-uniform minimal cycle of length ` is a cyclic list of k-setsA1,A2, . . . ,A` such that consecutive sets intersect in at leastone element and nonconsecutive sets are disjoint. The set of

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

A k-uniform linear cycle of length `, denoted by C(k)` is a

cyclic list of k-sets A1,A2, . . . ,A` such that consecutive setsintersect in exactly one element and nonconsecutive sets aredisjoint.


3. Matching Numbers

Theorem (Furedi and Jiang)

Let t b e a positive integer, k ≥ 4. For sufficiently larger n, wehave

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 ofall the k-sets in [n] that meet some fixed k-set S plus oneadditional k-set outside S.

Z. Furedi and T. Jiang, Hypergraph turan numbers of linearcycles, arXiv:1302.2387.[math.CO]2013.


3. Matching Numbers

Theorem (Furedi and Jiang)

Let t b e a positive integer, k ≥ 5. For sufficiently larger n, wehave

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 plusall the k-sets in [n] \ S that contain some two two fixed elements.

Z. Furedi and T. Jiang, Hypergraph turan numbers of linearcycles, arXiv:1302.2387.[math.CO]2013.


3. Matching Numbers

S = {C1 ∪ . . . ∪ Cr : Ci ∈ C(k)ì

for i ∈ [r ]}

Theorem (Gu, Li and Shi)

Let integers k ≥ 4, r ≥ 1, `1, . . . , `r ≥ 3, t =r∑

i=1

bì + 1

2c − 1, and

I = 1 if all the `1, . . . , `r are even, I = 0 otherwise. For sufficientlylarge n,

exk(n;S(`1, . . . , `r )) =

(n

k

)−(

n − t

k

)+ I .

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


3. Matching Numbers

Theorem (Gu, Li and Shi)

Let integers k ≥ 5, r ≥ 1, `1, . . . , `r ≥ 3, t =r∑

i=1

bì + 1

2c − 1, and

J =(n−t−2

k

)if all the `1, . . . , `r are even, J = 0 otherwise. For

sufficiently large n,

exk(n;C(k)`1, . . . ,C(k)

`r) =

(n

k

)−(

n − t

k

)+ J.

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


3. Matching Numbers

Bkn(q) = {(A, f ) : A ∈([n]k

)}.

Theorem

For positive integers q, n and k with k ≤ n, if F ⊂ Bkn(q) withν(F) = s where s < q, then

|F| ≤ sqk−1

(n − 1

k − 1

),

and equality holds if and only if F is isomorphic toF1 = {(A, f ) ∈ Bkn(q) : 1 ∈ A, f (1) ∈ [s]}.


3. Matching Numbers

Bkn(q) = {(A, f ) : A ∈([n]k

)}.

Theorem

For positive integers q, n and k with k ≤ n, if F ⊂ Bkn(q) withν(F) = s where s < q, then

|F| ≤ sqk−1

(n − 1

k − 1

),

and equality holds if and only if F is isomorphic toF1 = {(A, f ) ∈ Bkn(q) : 1 ∈ A, f (1) ∈ [s]}.


3. Matching Numbers

Let V be a finite set and let p be an ideal of 2V , that is, pconsists of subsets of V such that B ∈ p if B ⊆ A for someA ∈ 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 ⊆ Bkn(q) : ν(F) ≤ s}.


3. Matching Numbers



M(V ) = {A ∈ p : |A| = αp(V )}


p = {F ⊆ Bkn(q) : ν(F) ≤ s}.


3. Matching Numbers



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 ⊆ Bkn(q) : ν(F) ≤ s}.


3. Matching Numbers



M(V ) = {A ∈ p : |A| = αp(V )}


p = {F ⊆ Bkn(q) : ν(F) ≤ s}.


3. Matching Numbers

To complete the proof, we only need to determine αp(Bkn (q)) and

M(Bkn (q)).


3. Matching Numbers

Lemma

Let V be a finite set and p an idea on 2V . Suppose that there is atransitive 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 eachS ∈M(V ).


3. Matching Numbers

Bkn(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 . . . n1 1 1 . . . 1 2 2 2 . . . 2 . . . q q q . . . q

(A, f ) is said to be contained in the above cycle if it consistsof k consecutive elements.

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

Determine αp(H).


3. Matching Numbers

Bkn(q) = {(A, f ) : A ∈([n]k


1 2 3 . . . n 1 2 3 . . . n . . . 1 2 3 . . . n1 1 1 . . . 1 2 2 2 . . . 2 . . . q q q . . . q



Determine αp(H).


3. Matching Numbers

Bkn(q) = {(A, f ) : A ∈([n]k


1 2 3 . . . n 1 2 3 . . . n . . . 1 2 3 . . . n1 1 1 . . . 1 2 2 2 . . . 2 . . . q q q . . . q



Determine αp(H).


3. Matching Numbers

Bkn(q) = {(A, f ) : A ∈([n]k


1 2 3 . . . n 1 2 3 . . . n . . . 1 2 3 . . . n1 1 1 . . . 1 2 2 2 . . . 2 . . . q q q . . . q



Determine αp(H).


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 pairwiseadjacent vertices in G . For S ⊆ V (G ), ω(G [S ]) written asω(S).

F of H with ν(F) = s iff ω(F) = s.

To determine αp(H) is equivalent to determine ω(G [H]).


3. Matching Numbers







3. Matching Numbers







3. Matching Numbers







3. Matching Numbers







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 kconsecutive elements, viewed cyclically.

Lemma

Let n, s and k be three integers with n ≥ k(s + 1). For any vertexsubset F of Kn:k with ω(F ) = s, then |F | ≤ ks. Moreover, ifn > k(s + 1), the equality holds if and only if F is the union of smaximum independent sets of Kn:k .


3. Matching Numbers



Lemma



3. Matching Numbers



Lemma



3. Matching Numbers

ω(G [H]) = ks

Let F be a family of Bkn(q) with ν(F) = s. Then by the formerlemmas we have

|F||Bkn(q)|

≤ sk

nq.

Therefore,

|F| ≤ sk

nqqk

(n

k

)= sqk−1

(n − 1

k − 1

).


3. Matching Numbers

For n positive numbers p1, p2, . . . , pn with p1 ≤ p2 ≤ · · · ≤ pn, letLp 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 ifF = {(i1, i2, . . . , in) ∈ Ln,p : ij ∈ S} for one s-subset S of [p1] andone j of [n] with pj = p1.


3. Matching Numbers

Theorem

Let n and s be two positive integers with s ≤ n. If F is a family ofSn 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−1r−1

)(n − 1)!

(n − r)!.


3. Matching Numbers

Theorem

Let n and s be two positive integers with s ≤ n. If F is a family ofSn 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−1r−1

)(n − 1)!

(n − r)!.


Many Thanks!


Date post:	19-May-2020
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Some Problems and Generalizations on Erdos-Ko-Rado …1. Erd}os-Ko-Rado Theorem Theorem (EKR Theorem...

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