The Boolean lattice consists of all subsets of partially ordered under the containment relation. Sperner's Theorem states that the largest antichain of the Boolean lattice is given by a middle layer: the collection of all sets of size , or also, if is odd, the collection of all sets of size . Given , choose each subset of with probability independently. We show that for every constant , the largest antichain among these subsets is also given by a middle layer, with probability tending to 1 as tends to infinity. This is best possible, and we also characterize the largest antichains for every constant . Our proof is based on some new variations of Sapozhenko's graph container method.
Antichain codes
A family of sets is said to be an antichain if for all distinct , and it is said to be a distance‐ code if every pair of distinct elements of has Hamming distance at least . Here, we prove that if is both an antichain and a distance‐ code, then . This result, which is best‐possible up to the implied constant, is a purely combinatorial strengthening of a number of results in Littlewood–Offord theory; for example, our result gives a short combinatorial proof of Hálasz's theorem, while all previously known proofs of this result are Fourier‐analytic.
more » « less- Award ID(s):
- 2103154
- PAR ID:
- 10479488
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 55
- Issue:
- 6
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 3053-3062
- Size(s):
- p. 3053-3062
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
ABSTRACT -
Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are far from all codewords, by probing a given word only at a very small (sublinear, typically constant) number of locations. Such codes form the combinatorial backbone of PCPs. A major open problem is whether there exist LTCs with positive rate, constant relative distance and testable with a constant number of queries. In this paper, we present a new approach towards constructing such LTCs using the machinery of high-dimensional expanders. To this end, we consider the Tanner representation of a code, which is specified by a graph and a base code. Informally, our result states that if this graph is part of an {\em agreement expander} then the local testability of the code follows from the local testability of the base code. Agreement expanders allow one to stitch together many mostly-consistent local functions into a single global function. High-dimensional expanders are known to yield agreement expanders with constant degree. This work unifies and generalizes the known results on testability of the Hadamard, Reed-Muller and lifted codes, all of which are proved via a single round of local self-correction: the corrected value at a vertex v depends on the values of all vertices that share a constraint with v. In the above codes this set includes all of the vertices. In contrast, in our setting the degree of a vertex might be a constant, so we cannot hope for one-round self-correction. We overcome this technical hurdle by performing iterative self correction with logarithmically many rounds and tightly controlling the error in each iteration using properties of the agreement expander. Given this result, the missing ingredient towards constructing a constant-query LTC with positive rate and constant relative distance is an instantiation of a base code and a constant-degree agreement expander that interact well with each other.more » « less
-
A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. We present a complete classification of all connected edge-transitive graphs on less than or equal to 20 vertices. We investigate biregular bipartite edge-transitive graphs and present connections to combinatorial designs, and we show that the Cartesian products of complements of complete graphs give an additional family of edge-transitive graphs.more » « less
-
Consider a binary linear code of length N, minimum distance dmin, transmission over the binary erasure channel with parameter 0 < < 1 or the binary symmetric channel with parameter 0 < < 1 2 , and block-MAP decoding. It was shown by Tillich and Zemor that in this case the error probability of the block-MAP decoder transitions “quickly” from δ to 1−δ for any δ > 0 if the minimum distance is large. In particular the width of the transition is of order O(1/ √ dmin). We strengthen this result by showing that under suitable conditions on the weight distribution of the code, the transition width can be as small as Θ(1/N 1 2 −κ ), for any κ > 0, even if the minimum distance of the code is not linear. This condition applies e.g., to Reed-Mueller codes. Since Θ(1/N 1 2 ) is the smallest transition possible for any code, we speak of “almost” optimal scaling. We emphasize that the width of the transition says nothing about the location of the transition. Therefore this result has no bearing on whether a code is capacity-achieving or not. As a second contribution, we present a new estimate on the derivative of the EXIT function, the proof of which is based on the Blowing-Up Lemma.more » « less
-
null (Ed.)The edit distance between two strings is defined as the smallest number of insertions , deletions , and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is “one of the biggest unsolved problems in the field of combinatorial pattern matching” [37]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an quantum algorithm that approximates the edit distance within a factor of 3. We further extend this result to an quantum algorithm that approximates the edit distance within a larger constant factor. Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to metric estimation and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of , with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds.more » « less
