The Rank-Ramsey Problem and the Log-Rank Conjecture
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| Format: | Preprint |
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2024
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| author | Beniamini, Gal Linial, Nati Shraibman, Adi |
| author_facet | Beniamini, Gal Linial, Nati Shraibman, Adi |
| contents | A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory.
We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erdős-Rényi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n^{2/3})$. A key component of this construction is our matrix-theoretic view of lifts.
We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Rank-Ramsey Problem and the Log-Rank Conjecture Beniamini, Gal Linial, Nati Shraibman, Adi Combinatorics Computational Complexity A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory. We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erdős-Rényi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n^{2/3})$. A key component of this construction is our matrix-theoretic view of lifts. We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey. |
| title | The Rank-Ramsey Problem and the Log-Rank Conjecture |
| topic | Combinatorics Computational Complexity |
| url | https://arxiv.org/abs/2405.07337 |