A distribution invariant for association schemes and strongly regular graphs

Bier, Thomas

doi:10.1016/0024-3795(84)90180-0

Cited by 22 publications

(18 citation statements)

References 6 publications

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“…Although Conjecture 1.1 and the Erdős-Ko-Rado theorem share the same bound and extremal example, there is no obvious way to translate one question into the other. Conjecture 1.1 has attracted a lot of attention due to its connections with the Erdős-Ko-Rado theorem [1,2,4,6,7,8,9,10,11,12,15,18,19,22,23,24,25,26,27], but still remains open. For more than two decades, Conjecture 1.1 was known to hold only when k|n [24] or when n is at least an exponential function of k [6,8,23,27].…”

Section: Introductionmentioning

confidence: 99%

The Manickam–Miklós–Singhi conjectures for sets and vector spaces

Chowdhury

Sarkis

Shahriari

2014

Journal of Combinatorial Theory, Series A

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More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers n, k with n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 k-element subsets whose sum is also nonnegative. We verify this conjecture when n ≥ 8k 2 , which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when k < 10 45 .Moreover, our arguments resolve the vector space analogue of this conjecture. Let V be an n-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V so that the sum of all weights is zero. Define the weight of a subspace S ⊂ V to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if n ≥ 3k, then the number of k-dimensional subspaces in V with nonnegative weight is at least the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.

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Section: Introductionmentioning

confidence: 99%

The Manickam–Miklós–Singhi conjectures for sets and vector spaces

Chowdhury

Sarkis

Shahriari

2014

Journal of Combinatorial Theory, Series A

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“…In other words, when can we guarantee that A(n, k) = n−1 k−1 ? This question was first raised by Bier and Manickam [4,5] in their study of the so-called first distribution invariant of the Johnson scheme. In 1987, Manickam and Miklós [15] proposed the following conjecture, which in the language of the Johnson scheme was also posed by Manickam and Singhi [16] in 1988.…”

Section: Introductionmentioning

confidence: 99%

Nonnegative k-sums, fractional covers, and probability of small deviations

Alon

Huang

Sudakov

2012

Journal of Combinatorial Theory, Series B

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More than twenty years ago, Manickam, Miklós, and Singhi conjectured that for any integers n, k satisfying n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 kelement subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for n ≥ 33k 2 . This substantially improves the best previously known exponential lower bound n ≥ e ck log log k . In addition we prove a tight stability result showing that for every k and all sufficiently large n, every set of n reals with a nonnegative sum that does not contain a member whose sum with any other k − 1 members is nonnegative, contains at least n−1 k−1 + n−k−1 k−1 − 1 subsets of cardinality k with nonnegative sum.

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“…In 1988 Bier and Delsarte introduced the i-th distribution invariant of an association scheme [2,3]; this concept was essential in the development of the MMS conjecture. In this paper we consider a version of the MMS conjecture for partial geometries.…”

Section: Introductionmentioning

confidence: 99%