High-Girth Steiner Triple Systems

Kwan, Matthew; Sah, Ashwin; Sawhney, Mehtaab; Simkin, Michael

doi:10.48550/arxiv.2201.04554

Cited by 7 publications

(40 citation statements)

References 26 publications

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“…Given g ∈ N, there is N 1.4 (g) ∈ N such that if N ≥ N 1.4 (g), then there exists a Latin square with girth greater than g. Theorem 1.4 can be proved in essentially the same way as the analogous theorem for Steiner triple systems in [33]. The only real complication concerns a "triangle-regularisation" lemma, which is much simpler in the setting of [33] than in the setting of Theorem 1.4. Basically, we need a fractional triangle-decomposition result for quasirandom tripartite graphs.…”

Section: Introductionmentioning

confidence: 85%

“…That is to say, intercalates are the "hardest" type of Latin subsquare to avoid. In fact, it seems plausible that the random construction used to prove Theorem 1.4 may have property N ∞ with probability Ω(1), but an attempt to prove this would require deconstructing the proof of [33,Theorem 1.1] to a far greater extent than we do in this paper.…”

Section: Further Directionsmentioning

confidence: 98%

“…Finally, in Section 8 we explain how to adapt our work in [33] to prove Theorems 1.1 and 1.4. This section is mostly targeted towards readers who have read [33] or at least its proof outline, though we do provide a high-level summary of the overall approach.…”

Section: Further Directionsmentioning

confidence: 99%

“…Recall that the girth of a graph is the length of its shortest cycle. In [38], Linial defines a cycle 5 in a Latin square L to be a set of rows A, a set of columns B, and a set of symbols C, with |A| + |B| + |C| > 3, such that the A × B subarray of L contains at least |A| + |B| + |C| − 2 symbols in the set C. He defines the girth of a Latin square L to be the minimum of |A| + |B| + |C| over all such cycles in L. These definitions are motivated by the Brown-Erdős-Sós problem in extremal hypergraph theory, and in particular by an old conjecture of Erdős on the existence of high-girth Steiner triple systems, which we recently proved in [33] (see [33] for definitions and motivation). Answering a conjecture of Linial [38], we show that there exist Latin squares with arbitrarily high girth.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.1 is closely related to Theorem 1.4: it is not hard to see that a Latin square has girth greater than 6 if and only if it has no intercalate. The methods in [33] (which we explain in this paper how to adapt to prove Theorem 1.4) easily yield a lower bound on the number of Latin squares or Steiner triple systems with girth greater than a given constant g (see [33,Theorem 1.3]), so with a simple calculation one can deduce Theorem 1.1 from the proof of Theorem 1.4 1.3. Cuboctahedra.…”

Section: Introductionmentioning

confidence: 99%

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Substructures in Latin squares

Kwan¹,

Sah²,

Sawhney³

et al. 2022

Preprint

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We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order-n Latin squares with no intercalate (i.e., no 2 × 2 Latin subsquare) is at least EN , where N is the number of intercalates in a uniformly random order-n Latin square.In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general large-deviation problem for intercalates in random Latin squares, up to constant factors in the exponent: for any constant 0 < δ ≤ 1 we have Pr[N ≤ (1 − δ)EN] = exp(−Θ(n 2 )) and for any constant δ > 0 we haveFinally, we show that in almost all order-n Latin squares, the number of cuboctahedra (i.e., the number of pairs of possibly degenerate 2 × 2 subsquares with the same arrangement of symbols) is of order n 4 , which is the minimum possible. As observed by Gowers and Long, this number can be interpreted as measuring "how associative" the quasigroup associated with the Latin square is.

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Section: Introductionmentioning

confidence: 85%

Section: Further Directionsmentioning

confidence: 98%

Section: Further Directionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Substructures in Latin squares

Kwan¹,

Sah²,

Sawhney³

et al. 2022

Preprint

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Conflict‐free hypergraph matchings

Glock,

Joos,

Kim

et al. 2024

Journal of London Math Soc

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A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost‐regular, uniform hypergraph with small maximum codegree has an almost‐perfect matching. We extend this result by obtaining a conflict‐free matching, where conflicts are encoded via a collection of subsets . We say that a matching is conflict‐free if does not contain an element of as a subset. Under natural assumptions on , we prove that has a conflict‐free, almost‐perfect matching. This has many applications, one of which yields new asymptotic results for so‐called ‘high‐girth’ Steiner systems. Our main tool is a random greedy algorithm which we call the ‘conflict‐free matching process’.

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“Less” Strong Chromatic Indices and the (7, 4)-Conjecture

Gyárfás,

Sárközy

2023

SScMath

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A proper edge coloring of a graph 𝐺 is strong if the union of any two color classes does not contain a path with three edges (i.e. the color classes are induced matchings). The strong chromatic index 𝑞(𝐺) is the smallest number of colors needed for a strong coloring of 𝐺. One form of the famous (6, 3)-theorem of Ruzsa and Szemerédi (solving the (6, 3)-conjecture of Brown–Erdős–Sós) states that 𝑞(𝐺) cannot be linear in 𝑛 for a graph 𝐺 with 𝑛 vertices and 𝑐𝑛2 edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞𝐴(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that the union of any two color classes does not contain a path or cycle with four edges, we call it an A-coloring. The second is 𝑞𝐵(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that all four-cycles are colored with four different colors, we call it a B-coloring. These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of 𝑞𝐴(𝐺), 𝑞𝐵(𝐺) where 𝐺 is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original special(7, 4)-conjecture. In order to understand the behavior of 𝑞𝐴(𝐺) and 𝑞𝐵(𝐺), we study these parameters for some graphs.We note that 𝑞𝐴(𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞𝐵(𝐺) is studied here for the first time.

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High-Girth Steiner Triple Systems

Abstract: We prove a 1973 conjecture due to Erdős on the existence of Steiner triple systems with arbitrarily high girth.

Cited by 7 publications

References 26 publications

Substructures in Latin squares

Substructures in Latin squares

Conflict‐free hypergraph matchings

“Less” Strong Chromatic Indices and the (7, 4)-Conjecture

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