The existence of designs

Keevash, Peter

doi:10.48550/arxiv.1401.3665

Cited by 129 publications

(245 citation statements)

References 30 publications

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“…For functions f = f (n) and g = g(n), we write f = O(g) to mean that there is a constant C such that |f | ≤ C|g|, f = Ω(g) to mean that there is a constant c > 0 such that f (n) ≥ c|g(n)| for sufficiently large n, f = Θ(g) to mean that that f = O(g) and f = Ω(g), and f = o(g) to mean that f /g → 0 as n → ∞. Also, following [34], the notation…”

Section: Introductionmentioning

confidence: 99%

Large deviations in random Latin squares

Kwan¹,

Sah²,

Sawhney³

2021

Preprint

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In this note, we study large deviations of the number N of intercalates (2 × 2 combinatorial subsquares which are themselves Latin squares) in a random n × n Latin square. In particular, for constant δ > 0 we prove that Pr(N ≤ (1 − δ)n 2 /4) ≤ exp(−Ω(n 2 )) and Pr(N ≥ (1 + δ)n 2 /4) ≤ exp(−Ω(n 4/3 (log n) 2/3 )), both of which are sharp up to logarithmic factors in their exponents. As a consequence, we deduce that a typical order-n Latin square has (1 + o(1))n 2 /4 intercalates, matching a lower bound due to Kwan and Sudakov and resolving an old conjecture of McKay and Wanless. Kwan was supported by NSF grant DMS-1953990. Sah and Sawhney were supported by NSF Graduate Research Fellowship Program DGE-1745302.1 To see the analogy to permutation matrices, note that a Latin square can equivalently, and more symmetrically, be viewed as an n × n × n zero-one array such that every axis-aligned line sums to exactly 1.2 Jacobson and Matthews [30] and Pittenger [48] designed Markov chains that converge to the uniform distribution, but it is not known whether these Markov chains mix rapidly.

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Section: Introductionmentioning

confidence: 99%

Large deviations in random Latin squares

Kwan¹,

Sah²,

Sawhney³

2021

Preprint

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“…This is not the case in the toroidal problem. We wonder if the methods of randomized algebraic construction [10] or iterative absorption [8] might be more appropriate.…”

Section: Kmentioning

confidence: 99%

“…In recent years there have been several breakthroughs relating to the construction, enumeration, and analysis of designs. These include the Radhakrishnan entropy method [22], extended by Linial and Luria [15,16] to give upper bounds on the number of designs; the Rödl nibble [24] and random greedy algorithms [26], used to construct approximate designs; and completion methods, such as randomized algebraic constructions [10] and iterative absorption [8], used to complete approximate designs. We also mention the emerging limit theory of combinatorial designs [6,4] from which this paper draws inspiration.…”

Section: Introductionmentioning

confidence: 99%

The number of $n$-queens configurations

Simkin¹

2021

Preprint

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The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n×n board. We show that there exists a constant α = 1.942±3×10 −3 such that Q(n) = ((1 ± o(1)ne −α ) n . The constant α is characterized as the solution to a convex optimization problem in P([−1/2, 1/2] 2 ), the space of Borel probability measures on the square.The chief innovation is the introduction of limit objects for n-queens configurations, which we call queenons. These are a convex set in P([−1/2, 1/2] 2 ). We define an entropy function that counts the number of n-queens configurations that approximate a given queenon. The upper bound uses the entropy method. For the lower bound we describe a randomized algorithm that constructs a configuration near a prespecified queenon and whose entropy matches that found in the upper bound. The enumeration of n-queens configurations is then obtained by maximizing the (concave) entropy function in the space of queenons.Along the way we prove a large deviations principle for n-queens configurations that can be used to study their typical structure.

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“…Indeed, the problem of the existence of designs -one of the most fascinating questions of mathematics whose origins go back to the 19th century -can be phrased as a perfect (hyper)graph packing problem. This problem was solved only recently, first by Keevash [14], and independently by Glock, Lo, Kühn and Osthus [8].…”

Section: Introductionmentioning

confidence: 99%

“…For obtaining a corresponding exact result it remains to remove the Ω(n 2 ) term, which is hard. Indeed, this gap in difficulty between an approximate and an exact result is quite common in the area of packing and is best illustrated by the increase in difficulty needed to get from Rödl's proof [25] that approximate designs exist to the existence of designs [14,8]. Joos, Kim, Kühn and Osthus [12] proved that Ringel's conjecture holds exactly for large bounded degree trees.…”

Section: Introductionmentioning

confidence: 99%

The tree packing conjecture for trees of almost linear maximum degree

Allen¹,

Böttcher²,

Clemens³

et al. 2021

Preprint

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We prove that there is c > 0 such that for all sufficiently large n, if T1, . . . , Tn are any trees such that Ti has i vertices and maximum degree at most cn/ log n, then {T1, . . . , Tn} packs into Kn. Our main result actually allows to replace the host graph Kn by an arbitrary quasirandom graph, and to generalize from trees to graphs of bounded degeneracy that are rich in bare paths, contain some odd degree vertices, and only satisfy much less stringent restrictions on their number of vertices.

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The existence of designs

Cited by 129 publications

References 30 publications

Large deviations in random Latin squares

Large deviations in random Latin squares

The number of $n$-queens configurations

The tree packing conjecture for trees of almost linear maximum degree

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