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10
Ahlswede–Khachatrian theorem
, 2013
"... The Erdős–Ko–Rado theorem determines the largest µpmeasure of an intersecting family of sets. We consider the analogue of this theorem to tintersecting families (families in which any two sets have at least t elements in common), following Ahlswede and Khachatrian [1, 2]. We present a proof of th ..."
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The Erdős–Ko–Rado theorem determines the largest µpmeasure of an intersecting family of sets. We consider the analogue of this theorem to tintersecting families (families in which any two sets have at least t elements in common), following Ahlswede and Khachatrian [1, 2]. We present a proof
A new short proof of a theorem of Ahlswede and Khachatrian
, 2007
"... Ahlswede and Khachatrian [5] proved the following theorem, which answered a question of Frankl and Füredi [3]. Let 2 ≤ t + 1 ≤ k ≤ 2t + 1 and n ≥ (t + 1)(k − t + 1). Suppose that F is a family of ksubsets of an nset, every two of which have at least t common elements. If  ∩F ∈F F  < t, then  ..."
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Ahlswede and Khachatrian [5] proved the following theorem, which answered a question of Frankl and Füredi [3]. Let 2 ≤ t + 1 ≤ k ≤ 2t + 1 and n ≥ (t + 1)(k − t + 1). Suppose that F is a family of ksubsets of an nset, every two of which have at least t common elements. If  ∩F ∈F F  < t
In this paper we study large families of finite, binary sequences
"... Ahlswede, Khachatrian, Mauduit and A. Sárközy introduced the notion familycomplexity of families of binary sequences. They estimated the familycomplexity of a large family related to Legendre symbol introduced by Goubin, Mauduit and Sárközy. Here their result is improved, and apart from the cons ..."
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Ahlswede, Khachatrian, Mauduit and A. Sárközy introduced the notion familycomplexity of families of binary sequences. They estimated the familycomplexity of a large family related to Legendre symbol introduced by Goubin, Mauduit and Sárközy. Here their result is improved, and apart from
ASYMPTOTIC UPPER BOUNDS ON THE SHADES OF tINTERSECTING FAMILIES
, 2008
"... We examine the mshades of tintersecting families of ksubsets of [n], and conjecture on the optimal upper bound on their cardinalities. This conjecture extends Frankl’s General Conjecture that was proven true by Ahlswede–Khachatrian. From this we deduce the precise asymptotic upper bounds on the ..."
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Cited by 2 (1 self)
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We examine the mshades of tintersecting families of ksubsets of [n], and conjecture on the optimal upper bound on their cardinalities. This conjecture extends Frankl’s General Conjecture that was proven true by Ahlswede–Khachatrian. From this we deduce the precise asymptotic upper bounds
New Binary FixFree Codes with Kraft Sum 3/4
, 2002
"... Two sufficient conditions are given for the existence of binary fixfree codes. The results move closer to the AhlswedeBalkenholKhachatrian conjecture that Kraft sums of at most 3=4 suffice for the existence of fixfree codes. ..."
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Cited by 2 (0 self)
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Two sufficient conditions are given for the existence of binary fixfree codes. The results move closer to the AhlswedeBalkenholKhachatrian conjecture that Kraft sums of at most 3=4 suffice for the existence of fixfree codes.
KATONA’S INTERSECTION THEOREM: FOUR PROOFS
, 2001
"... It is known from a previous paper [3] that Katona’s Intersection Theorem follows from the Complete Intersection Theorem by Ahlswede and Khachatrian via a Comparison Lemma. It also has been proved directly in [3] by the pushing–pulling method of that paper. Here weaddathirdproofviaanew(k,k+1)shiftin ..."
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Cited by 3 (1 self)
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It is known from a previous paper [3] that Katona’s Intersection Theorem follows from the Complete Intersection Theorem by Ahlswede and Khachatrian via a Comparison Lemma. It also has been proved directly in [3] by the pushing–pulling method of that paper. Here weaddathirdproofviaanew(k,k+1
On the most Weight w Vectors in a Dimension k Binary Code
"... Ahlswede, Aydinian, and Khachatrian posed the following problem: what is the maximum number of Hamming weight w vectors in a kdimensional subspace of F n 2? The answer to this question could be relevant to coding theory, since it sheds light on the weight distributions of binary linear codes. We gi ..."
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Ahlswede, Aydinian, and Khachatrian posed the following problem: what is the maximum number of Hamming weight w vectors in a kdimensional subspace of F n 2? The answer to this question could be relevant to coding theory, since it sheds light on the weight distributions of binary linear codes. We
Sufficient Conditions for Existence of Binary FixFree Codes
"... Abstract—Two sufficient conditions are given for the existence of binary fixfree codes (i.e., both prefixfree and suffixfree). Let be a finite multiset of positive integers whose Kraft sum is at most Q R. It is shown that there exists a fixfree code whose codeword lengths are the elements of if ..."
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of if either of the following two conditions holds. i) The smallest integer in is at least P, and no integer in, except possibly the largest one, occurs more than P ��� @ A P times. ii) No integer in, except possibly the largest one, occurs more than twice. The results move closer to the Ahlswede–Balkenhol–Khachatrian
Old And New Results For The Weighted tIntersection Problem Via AKMethods
, 1998
"... . Let [n] := f1; : : : ; ng, 2 [n] be the power set of [n] and s 2 [n]. A family F ` 2 [n] is called tintersecting in [s] if jX 1 " X 2 " [s]j t for all X 1 ; X 2 2 F : Let ! : 2 [n] ! R+ be a given weight function and M s (n; t; !) := maxf!(F) : F is tintersecting in [s]g: For ..."
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Cited by 14 (2 self)
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]g: For several weight functions, the numbers Mn (n; t; !) can be determined using three important methods of Ahlswede and Khachatrian: Generating Sets [2], Comparison Lemma [4], and PushingPulling [3]. We survey these methods. Also, sufficient conditions on ! for the equality M s (n; t; !) = Mn (n; t
Pairwise intersections and forbidden configurations
, 2006
"... Let fm(a, b, c, d) denote the maximum size of a family F of subsets of an melement set for which there is no pair of subsets A, B ∈ F with A ∩ B  ≥a, Ā ∩ B  ≥b, A ∩ ¯B  ≥c, and Ā ∩ ¯B  ≥d. By symmetry we can assume a ≥ d and b ≥ c. We show that fm(a, b, c, d) is Θ(m a+b−1) if either b> c ..."
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Cited by 7 (5 self)
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key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.