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Ramsey and Turán numbers of sparse hypergraphs

Jacob Fox, Maya Sankar, Michael Simkin, Jonathan Tidor, Yunkun Zhou

TL;DR

This paper introduces skeletal degeneracy, notably the 1-skeletal degeneracy d_1(H), as a hypergraph analogue of graph degeneracy and shows it governs key extremal properties. Using dependent random choice and a random-greedy embedding approach, it proves upper and lower bounds on Turán numbers for k-uniform, k-partite hypergraphs that depend on d_1(H), and establishes linear (and almost-linear) Ramsey bounds for such families. A detailed pruning-embedding framework yields a linear r(H;q) bound, with a layered partitioning strategy for both H and the host graph, and a refined defect control mechanism. The concluding remarks posit that the maximal skeletal degeneracy d_{max}(H) more precisely determines Turán exponents, and present structural results that unify degeneracy control across all i-skeletons, together with conjectures and constructions (e.g., hedgehogs and Latin-square-inspired hypergraphs) illustrating the sharpness and limitations of these bounds.

Abstract

Degeneracy plays an important role in understanding Turán- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a $k$-uniform hypergraph as the degeneracy of its $1$-skeleton (i.e., the graph formed by replacing every $k$-edge by a $k$-clique). We prove that skeletal degeneracy controls hypergraph Turán and Ramsey numbers in a similar manner to (graphical) degeneracy. Specifically, we show that $k$-uniform hypergraphs with bounded skeletal degeneracy have linear Ramsey number. This is the hypergraph analogue of the Burr-Erdős conjecture (proved by Lee). In addition, we give upper and lower bounds of the same shape for the Turán number of a $k$-uniform $k$-partite hypergraph in terms of its skeletal degeneracy. The proofs of both results use the technique of dependent random choice. In addition, the proof of our Ramsey result uses the `random greedy process' introduced by Lee in his resolution of the Burr-Erdős conjecture.

Ramsey and Turán numbers of sparse hypergraphs

TL;DR

This paper introduces skeletal degeneracy, notably the 1-skeletal degeneracy d_1(H), as a hypergraph analogue of graph degeneracy and shows it governs key extremal properties. Using dependent random choice and a random-greedy embedding approach, it proves upper and lower bounds on Turán numbers for k-uniform, k-partite hypergraphs that depend on d_1(H), and establishes linear (and almost-linear) Ramsey bounds for such families. A detailed pruning-embedding framework yields a linear r(H;q) bound, with a layered partitioning strategy for both H and the host graph, and a refined defect control mechanism. The concluding remarks posit that the maximal skeletal degeneracy d_{max}(H) more precisely determines Turán exponents, and present structural results that unify degeneracy control across all i-skeletons, together with conjectures and constructions (e.g., hedgehogs and Latin-square-inspired hypergraphs) illustrating the sharpness and limitations of these bounds.

Abstract

Degeneracy plays an important role in understanding Turán- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a -uniform hypergraph as the degeneracy of its -skeleton (i.e., the graph formed by replacing every -edge by a -clique). We prove that skeletal degeneracy controls hypergraph Turán and Ramsey numbers in a similar manner to (graphical) degeneracy. Specifically, we show that -uniform hypergraphs with bounded skeletal degeneracy have linear Ramsey number. This is the hypergraph analogue of the Burr-Erdős conjecture (proved by Lee). In addition, we give upper and lower bounds of the same shape for the Turán number of a -uniform -partite hypergraph in terms of its skeletal degeneracy. The proofs of both results use the technique of dependent random choice. In addition, the proof of our Ramsey result uses the `random greedy process' introduced by Lee in his resolution of the Burr-Erdős conjecture.
Paper Structure (15 sections, 30 theorems, 118 equations, 1 algorithm)

This paper contains 15 sections, 30 theorems, 118 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Upper bounds on the Turán number
  3. Lower bounds on the Turán number
  4. Almost-linear Ramsey number
  5. Linear Ramsey number
  6. Pruning
  7. Embedding
  8. Putting it all together
  9. Partitioning $H$
  10. Partitioning $G$
  11. Finishing up
  12. Concluding remarks
  13. Better bounds on the Turán number
  14. Relations between various skeletal degeneracies
  15. More conjectures on max-skeletal degeneracy

Key Result

Theorem 1.1

For every $d$-degenerate bipartite graph $H$,

Theorems & Definitions (68)

  • Theorem 1.1: AKS03
  • Theorem 1.2: Lee17
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 58 more