Skip to main content

A new Alon-Babai-Suzuki-type inequality on set systems

Abstract

Let K={ k 1 , k 2 ,, k r } and L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers. Let be a family of subsets of [n] such that |F|K for every FF and |EF|L for every pair of distinct subsets E,FF. We prove that

|F| ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 s 2 r + 1 )

when we have the conditions that KL= and ns+max k i . This result gives an improvement for Alon, Babai, and Suzuki’s conjecture under the nonmodular. This paper gets an improvement of a theorem of Hwang et al. with KL=.

MSC:05D05.

1 Introduction

In this paper, let X denote the set [n]={1,2,,n} and L={ l 1 , l 2 ,, l s } be a set of s nonnegative integers. A family of subsets of X is called -intersecting if |EF|L for every pair of distinct subsets E,FF. A family is k-uniform if it is a collection of k-subsets of X.

In 1975, Ray-Chaudhuri and Wilson proved the following fundamental result.

Theorem 1.1 (Ray-Chaudhuri and Wilson [1])

If is a k-uniform -intersecting family of subsets of X, then

|F| ( n s ) .

In terms of the parameters n and s, this inequality is best possible, as shown by the set of all s-subsets of X with L={0,1,,s1}. A nonuniform version of Theorem 1.1 was proved by Frankl and Wilson in 1981.

Theorem 1.2 (Frankl and Wilson [2])

If is an -intersecting family of subsets of X, then

|F| ( n s ) + ( n s 1 ) ++ ( n 0 ) .

This result is again best possible in terms of the parameters n and s, as shown by the family of all subsets of size at most s of X with L={0,1,,s1}.

In 1991, Alon, Babai, and Suzuki proved the following inequality, which is a generalization of the well-known Frankl-Ray-Chaudhuri-Wilson theorems (Theorem 1.1 and Theorem 1.2).

Theorem 1.3 (Alon, Babai, and Suzuki [3])

Let K={ k 1 , k 2 ,, k r }, L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that |F|K for every FF. If min k i >sr, then

|F| ( n s ) + ( n s 1 ) ++ ( n s r + 1 ) .

Note that it is best possible in terms of the parameters n, r, and s, as shown by the set of all subsets of X with size at least sr+1 and at most s, and L={0,1,,s1}.

Since then, many Alon-Babai-Suzuki-type inequalities have been proved. Below is a list of related results in this field obtained by others.

Theorem 1.4 Let K={ k 1 , k 2 ,, k r }, L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that |F|K for every FF. If KL=, then

|F| ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 0 ) .

Theorem 1.5 (Snevily [4])

Let K={ k 1 , k 2 ,, k r }, L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that |F|K for every FF. If min k i >max l j , then

|F| ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 s 2 r + 1 ) .

Theorem 1.6 (Hwang and Sheikh [5])

Let K={k,k+1,,k+r1}, L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that |F|K for every FF. If KL= and k>sr, then

|F| ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 s r ) .

Theorem 1.7 (Hwang and Sheikh [5])

Let K={ k 1 , k 2 ,, k r }, L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that |F|K for every FF. If F F F and min k i >sr, then

|F| ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 s r ) .

Conjecture 1.8 (Alon, Babai, and Suzuki [3])

Let K and be subsets of {0,1,,p1} such that KL=, where p is a prime and F={ F 1 , F 2 ,, F m } be a family of subsets of [n] such that | F i |(modp)K for all F i F and | F i F j |(modp)L for ij. If ns+max k i for every i, then |F| ( n s ) + ( n s 1 ) ++ ( n s r + 1 ) .

In the following paper, they prove the above conjecture in the nonmodular.

Theorem 1.9 (Hwang et al. [6])

Let K={ k 1 , k 2 ,, k r }, L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that |F|K for every FF. If ns+max k i , then

|F| ( n s ) + ( n s 1 ) ++ ( n s r + 1 ) .

In this paper, we will prove the following Alon-Babai-Suzuki-type inequality which gives an improvement for Theorem 1.9 with KL= and for Alon, Babai and Suzuki’s conjecture in the nonmodular.

Theorem 1.10 Let K={ k 1 , k 2 ,, k r }, L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that |F|K for every FF. If KL= and ns+max k i , then

|F| ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 s 2 r + 1 ) .

We note that in some cases the conditions KL= and ns+max k i in Theorem 1.10 holds, but Snevily’s condition min k i >max l j in Theorem 1.5 does not. For instance, if n=12, K={3,6} and L={0,1,2,4,5}, then it is clear that KL= and s+max k i =5+6<12=n, but min k i =3<5=max l j .

If the condition KL= in Theorem 1.10 is replaced by F F F, we have the following result which gives a better bound than Theorem 1.10 and Theorem 1.7.

Theorem 1.11 Let K={ k 1 , k 2 ,, k r }, L={ l 1 , l 2 ,, l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that |F|K for every FF. If F F F and ns+max k i , then

|F| ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 s r + 1 ) .

2 Proofs of theorems

In this section, we will give a proof for Theorem 1.10, which is motivated by the methods used in [37].

Proof of Theorem 1.10 Let F={ F 1 , F 2 ,, F m }. We may assume (after relabeling) that for 1it, n F i and that for i>t, n F i . With each set F i F, we associate its characteristic vector v i =( v i 1 ,, v i n ) R n , where v i j =1 if j F i and v i j =0 otherwise. Let v i ¯ be the characteristic vector of F i c .

Recall that a polynomial in n variables is multilinear if its degree in each variable is at most 1. Let us restrict the domain of the polynomials we will work with to the n-cube Ω= { 0 , 1 } n R n . Since in this domain x i 2 = x i for each variable, every polynomial in our proof is multilinear.

For each F i F, define

f i (x)= l j L ( v i x ( | F i | l j ) ) .

Then f i ( v i ¯ )0 for every 1im and f i ( v j ¯ )=0 for ij, since KL=. Thus, { f i (x)|1im} is a linearly independent family.

Let G={ G 1 ,, G p } be the family of subsets of X with size at most s that contain n, which is ordered by size, that is, | G i || G j | if i<j, where p= i = 0 s 1 ( n 1 i ) . Let u i denote the characteristic vector of G i . For i=1,,p, we define

g i (x)= j G i x j .

Since g i ( u i )0 for every 1ip and g i ( u j )=0 for any j<i, { g i (x)|1ip} is a linearly independent family.

Let H={ H 1 ,, H q } be the family of subsets of [n]{n} with size at most s2r, where q= i = 0 s 2 r ( n 1 i ) . We order the members of such that | H i || H j | if i<j. Let w i be the characteristic vector of H i .

Let W={n k i 1| k i K}{n k i | k i K}. Then |W|2r. Set

f(x)= h W ( j = 1 n 1 x j h ) .

For i=1,,q, define

h i (x)=f(x) j H i x j .

Note that h i ( w j )=0 for any j>i and h i ( w i )0 for every 1iq since ns+max k i , and thus { h i (x)|1iq} is a linearly independent family.

We will show that the polynomials in

{ f i ( x ) | 1 i m } { g i ( x ) | 1 i p } { h i ( x ) | 1 i q }

are linearly independent. Suppose that we have a linear combination of these polynomials that equals zero:

i = 1 m α i f i (x)+ i = 1 p β i g i (x)+ i = 1 q γ i h i (x)=0.
(2.1)

We will prove that the coefficients must be zero. First substitute the characteristic vector v i ¯ of F i c with n F i into equation (2.1). Since n G j , g j ( v i ¯ )=0 for every 1jp. Note that we have h j ( v i ¯ )=0 for every 1jq. Recall that f j ( v i ¯ )=0 for ji, we obtain α i f i ( v i ¯ )=0. Thus, α i =0 for every 1it, since f i ( v i ¯ )0. It follows that

i = t + 1 m α i f i (x)+ i = 1 p β i g i (x)+ i = 1 q γ i h i (x)=0.
(2.2)

Then we substitute the characteristic vector v i ¯ of F i c {n} with n F i into equation (2.2). For every 1jq and 1kp, h j ( v i ¯ )=0, g k ( v i ¯ )=0. Note that f j ( v i ¯ )=0 for ji, we obtain α i f i ( v i ¯ )=0. Since f i ( v i ¯ )0, α i =0 for every t+1im. Therefore, equation (2.2) reduces to

i = 1 p β i g i (x)+ i = 1 q γ i h i (x)=0.
(2.3)

First, we substitute the characteristic vector w i of H i with the smallest size into equation (2.3). We follow the same process to substitute the characteristic vector w i of H i with the smallest size after deleting first H i . Note that h i ( w j )=0 for any j>i and h i ( w i )0 for every 1iq, since H i does not contain n and G i contains n, g k ( w i )=0 for 1kp. Thus, we reduce (2.3) to

i = 1 p β i g i (x)=0.
(2.4)

We prove that { g i (x)|1ip} is already a linearly independent family. To complete the proof, simply note that each polynomial in { f i (x)|1im}{ g i (x)|1ip}{ h i (x)|1iq} can be written as a linear combination of the multilinear polynomials of degree at most s. The space of such multilinear polynomials has dimension i = 0 s ( n i ) . It follows that

m+p+q=|F|+ i = 0 s 1 ( n 1 i ) + i = 0 s 2 r ( n 1 i ) i = 0 s ( n i )

which implies

|F| ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 s 2 r + 1 ) .

This completes the proof of the theorem. □

Proof of Theorem 1.11 Let n F F F. Then consider F ={ F 1 , F 2 ,, F m } where F i = F i {n} for 1im. Now, | F i | K ={ k 1 , k 2 ,, k r }, where k i = k i 1. Similarly, | F i F j | L ={ l 1 , l 2 ,, l s }, where l i = l i 1. Since ns+max k i , then n1s+max k i . Thus, it follows from Theorem 1.9 that

m ( n 1 s ) + ( n 1 s 1 ) ++ ( n 1 s r + 1 ) .

 □

References

  1. Ray-Chaudhuri DK, Wilson RM: On t -designs. Osaka J. Math. 1975, 12: 737–744.

    MathSciNet  Google Scholar 

  2. Frankl P, Wilson RM: Intersection theorems with geometric consequences. Combinatorica 1981, 1: 357–368. 10.1007/BF02579457

    MathSciNet  Article  Google Scholar 

  3. Alon N, Babai L, Suzuki H: Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson-Type intersection theorems. J. Comb. Theory, Ser. A 1991, 58: 165–180. 10.1016/0097-3165(91)90058-O

    MathSciNet  Article  Google Scholar 

  4. Snevily HS: A generalization of the Ray-Chaudhuri-Wilson theorem. J. Comb. Des. 1995, 3: 349–352. 10.1002/jcd.3180030505

    MathSciNet  Article  Google Scholar 

  5. Hwang K-W, Sheikh NN: Intersection families and Snevily’s conjecture. Eur. J. Comb. 2007, 28: 843–847. 10.1016/j.ejc.2005.11.002

    MathSciNet  Article  Google Scholar 

  6. Hwang K-W, Kim T, Jang LC, Kim P, Sohn G: Alon-Babai-Suzuki’s conjecture related to binary codes in nonmodular version. J. Inequal. Appl. 2010., 2010: Article ID 546015. doi:10.1155/2010/546015

    Google Scholar 

  7. Snevily HS: On generalizations of the de Bruijn-Erdös theorem. J. Comb. Theory, Ser. A 1994, 68: 232–238. 10.1016/0097-3165(94)90103-1

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Dong-A University research fund.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kyung-Won Hwang.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

KWH made the first polynomial and proved the first polynomials are linear independent. RXJL made the other polynomials and proved that the other polynomials are linearly independent. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Liu, R.X., Hwang, KW. A new Alon-Babai-Suzuki-type inequality on set systems. J Inequal Appl 2013, 171 (2013). https://doi.org/10.1186/1029-242X-2013-171

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-171

Keywords

  • Alon-Babai-Suzuki inequalities
  • Frankl-Ray-Chaudhuri-Wilson theorems
  • multilinear polynomials