A lower bound for the k‐multicolored sum‐free problem in Zmn

Lovász, László; Sauermann, Lisa

doi:10.1112/plms.12223

Cited by 11 publications

(7 citation statements)

References 37 publications

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“…This was the main ingredient in the recent cap-set problem breakthrough [9] and the driving force behind many recent developments in additive combinatorics. We refer the reader to [6] for the original application for which it was developed and to [22] for a better account of its recent history and a comprehensive list of references.…”

Section: An Algebraic Proof Of Kleitman's Theoremmentioning

confidence: 99%

On subsets of the hypercube with prescribed Hamming distances

Huang

Klurman

Pohoata

2020

Journal of Combinatorial Theory, Series A

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A theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in {0, 1} n with diameter d has cardinality at most that of a Hamming ball of radius d/2. In this paper, we give an algebraic proof of Kleitman's Theorem, by carefully choosing a pseudo-adjacency matrix for certain Hamming graphs, and applying the Cvetković bound on independence numbers. This method also allows us to prove several extensions and generalizations of Kleitman's Theorem to other allowed distance sets, in particular blocks of consecutive integers that do not necessarily grow linearly with n. We also improve on a theorem of Alon about subsets of F n p whose difference set does not intersect {0, 1} n nontrivially.2 n−1 0 + · · · + n−1 t , for d = 2t + 1.. Both inequalities are sharp. When d = 2t, the upper bound is attained by a Hamming ball of radius d/2, for instance, a collection of binary vectors having at most t 1-coordinates. For d = 2t + 1, an optimal example is given by the Cartesian product of {0, 1} and the (n − 1)dimensional Hamming ball of radius t, or alternatively, a Hamming ball of radius d/2 centered at (1/2, 0, · · · , 0). Kleitman's Theorem can be reduced to the celebrated Katona intersection theorem [18], using the squashing operation (see [2] or [14]). Generalizations of this theorem have been studied for the n-dimensional grid [m] n with Hamming distance [3,13]; as well as [m] n and the n-dimensional torus Z n m with Manhattan distance [1,5,8].

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Section: An Algebraic Proof Of Kleitman's Theoremmentioning

confidence: 99%

On subsets of the hypercube with prescribed Hamming distances

Huang

Klurman

Pohoata

2020

Journal of Combinatorial Theory, Series A

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“…A priori, for Φ ⊆ V 1 ×• • •×V k we have the upper bound Q(Φ) min i |V i | and therefore it holds that Q(Φ) min i |V i |, since |V ×n i | = |V i | n . Problem 1 has been studied for several families of k-graphs, in several different contexts: the cap set problem [12,33,19,23,24], approaches to fast matrix multiplication [32,4,5,28], arithmetic removal lemmas [21,14], property testing [15,17], quantum information theory [35,36], and the general study of asymptotic properties of tensors [34,7,8]. We finally mention the related result of Ruzsa and Szemerédi which says that the largest subset E ⊆ n 2 such that (E×E×E)∩{({a, b}, {b, c}, {c, a}) : a, b, c ∈ [n]} is a matching, has size n 2−o (1) |E| o(n 2 ) when n goes to infinity [27], see also [2, Equation 2].…”

Section: Asymptotic Induced Matchingsmentioning

confidence: 99%

The Asymptotic Induced Matching Number of Hypergraphs: Balanced Binary Strings

Arunachalam

Vrana

Zuiddam³

2020

Electron. J. Combin.

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We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith–Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science.Our proof relies on a new combinatorial inequality that may be of independent interest. This inequality concerns how many pairs of Boolean vectors of fixed Hamming weight can have their sum in a fixed subspace.

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“…If G = F n p , then Tao's slice rank formulation [21] of the Croot-Lev-Pach polynomial method [6] yields an upper bound for the size of a k-colored sum-free set in F n p for any k ≥ 3 (and if p and k are fixed and n is large, then this bound is essentially tight [17]). To state this bound, let us define…”

Section: Proof Overviewmentioning

confidence: 99%

On the size of subsets of $\mathbb{F}_p^{n}$ without $p$ distinct elements summing to zero

Sauermann¹

2019

Preprint

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Let us fix a prime p. The Erdős-Ginzburg-Ziv problem asks for the minimum integer s such that any collection of s points in the lattice Z n contains p points whose centroid is also a lattice point in Z n . For large n, this is essentially equivalent to asking for the maximum size of a subset of F n p without p distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime p ≥ 5 and large n. In particular, we prove that any subset of F n p without p distinct elements summing to zero has size at most Cp • 2 √ p n , where Cp is a constant only depending on p. For p and n going to infinity, our bound is of the form p (1/2)•(1+o(1))n , whereas all previously known upper bounds were of the form p (1−o(1))n (with p n being a trivial bound).Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds.

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A lower bound for the k‐multicolored sum‐free problem in Zmn

Cited by 11 publications

References 37 publications

On subsets of the hypercube with prescribed Hamming distances

On subsets of the hypercube with prescribed Hamming distances

The Asymptotic Induced Matching Number of Hypergraphs: Balanced Binary Strings

On the size of subsets of $\mathbb{F}_p^{n}$ without $p$ distinct elements summing to zero

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