Flattening rank and its combinatorial applications

One of the central questions in Ramsey theory asks how small the largest clique and independent set in a graph on N vertices can be. By the celebrated result of Erdős from 1947, a random graph on N vertices with edge probability $1/2$ contains no clique or independent set larger than $2\log _2 N$ , with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools. Say that an r-uniform hypergraph $\mathcal {H}$ is algebraic of complexity $(n,d,m)$ if the vertices of $\mathcal {H}$ are elements of $\mathbb {F}^{n}$ for some field $\mathbb {F}$ , and there exist m polynomials $f_1,\dots ,f_m:(\mathbb {F}^{n})^{r}\rightarrow \mathbb {F}$ of degree at most d such that the edges of $\mathcal {H}$ are determined by the zero-patterns of $f_1,\dots ,f_m$ . The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity $(n,d,m)$ has good Ramsey properties, then at least one of the parameters $n,d,m$ must be large. In 2001, Rónyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if G is an algebraic graph of complexity $(n,d,m)$ on N vertices, then either G or its complement contains a complete balanced bipartite graph of size $\Omega _{n,d,m}(N^{1/(n+1)})$ . We extend this result by showing that such G contains either a clique or an independent set of size $N^{\Omega (1/ndm)}$ and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for r-uniform algebraic hypergraphs that are defined by a single polynomial that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.

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“…, 𝑎 𝑟 ). We prove the following result in the companion note [34]. Let N 𝑟 ,𝑠 be the family of r-uniform hypergraphs N having the following form.…”

Section: Regularity Lemma For Algebraic Hypergraphsmentioning

confidence: 99%

“…A detailed discussion of i-flattening rank and its combinatorial applications can be found in [34]. In this paper, we use the following property of the i-flattening rank of tensors defined by polynomials.…”

Section: Linear Algebramentioning

confidence: 99%

Ramsey properties of algebraic graphs and hypergraphs

Sudakov

Tomon

2022

Forum of Mathematics, Sigma

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“…In words: 2n − 1 matchings of size n in a bipartite graph have a rainbow matching of size n. A very nice recent result, using algebra for its proof, is: Theorem 2.5. [27] Let C r be the class of r-uniform hypergraphs. → Cr t.…”

Section: A General Theme -Big In One Sense Small In Anothermentioning

confidence: 99%

Choice Functions

Aharoni¹,

Briggs²

2021

Preprint

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This is a survey paper on rainbow sets (another name for "choice functions").The main theme is the distinction between two types of choice functions: those having a large (in the sense of belonging to some specified filter, namely closed up set of sets) image, and those that have a large domain and small image, where "smallness" means belonging to some specified complex (a closed-down set). The paper contains some new results:(1) theorems on scrambled versions, in which the sets are re-shuffled before choosing the rainbow set, and (2) results on weighted and cooperative versions -to be defined below.

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Choice Functions

Aharoni

Briggs

2022

Order

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Flattening rank and its combinatorial applications

Cited by 5 publications

References 9 publications

Ramsey properties of algebraic graphs and hypergraphs

Ramsey properties of algebraic graphs and hypergraphs

Choice Functions

Choice Functions

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