A double-exponential lower bound for $r_4(5,n)$

Abstract: The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. We prove that $r_4(5,n)\ge 2^{2^{cn^{1/7}}}$, where $c>0$ is an absolute constant. As a consequence, we determine the tower growth rate of $r_k(k+1,n)$, which completely solves the problem of establishing the tower growth rate for all classical off-diagonal hypergraph Ramsey numbers, first posed by Erdős and Hajnal in 1972.

• The paper proves a double-exponential lower bound for r₄(5,n), definitively settling the tower growth rate for off-diagonal hypergraph Ramsey numbers.
• It employs a combination of probabilistic coloring and a refined multi-layer stepping-up method to construct hypergraphs that exclude forbidden substructures.
• The results confirm longstanding conjectures by Erdős and Hajnal, setting a benchmark for the complexity of classical off-diagonal hypergraph Ramsey numbers.

Double-Exponential Lower Bound for $r_4(5, n)$: Determining Tower Growth in Hypergraph Ramsey Numbers

Introduction

The paper "A double-exponential lower bound for $r_4(5,n)$" (2604.23986) addresses a longstanding problem in extremal combinatorics concerning off-diagonal Ramsey numbers for $k$-uniform hypergraphs. In particular, it proves a double-exponential lower bound for the four-uniform hypergraph Ramsey number $r_4(5,n)$, thus completely resolving the tower growth rate for all classical off-diagonal hypergraph Ramsey numbers $r_k(k+1,n)$, a question first posed by Erdős and Hajnal in 1972. This result constitutes a decisive step in understanding the extremal function growth for hypergraph Ramsey numbers outside the diagonal regime.

Background and Prior Work

For integers $k\ge 2$, $s>k$, and $n$, $r_k(s, n)$ denotes the minimum $N$ such that every $r_4(5,n)$0-vertex $r_4(5,n)$1-graph contains either a $r_4(5,n)$2 (a clique of size $r_4(5,n)$3) or an independent set of size $r_4(5,n)$4. Off-diagonal Ramsey numbers ($r_4(5,n)$5, $r_4(5,n)$6) have historically exhibited tower-type growth with regards to $r_4(5,n)$7 as $r_4(5,n)$8 and $r_4(5,n)$9 are fixed and $k$0 increases.

Previous studies established upper bounds on $k$1 using the classical Erdős-Hajnal stepping-up lemma, yielding $k$2, where $k$3 denotes a $k$4-level tower function. Lower bounds, however, generally trailed behind, particularly for $k$5. Mubayi and Suk [M-S-3] showed that $k$6, yet the existence of a tower-type lower bound matching the upper remained elusive for all $k$7. The case $k$8, $k$9 was recognized as fundamental for resolving this gap.

Main Results

The central result of the paper is the establishment of a double-exponential lower bound for $r_4(5,n)$0:

$r_4(5,n)$1

with $r_4(5,n)$2 an absolute constant. This lower bound is proven via probabilistic and combinatorial construction, combining a coloring argument and a refined application of the stepping-up technique.

As a direct consequence, the authors derive that $r_4(5,n)$3 for $r_4(5,n)$4 and some constant $r_4(5,n)$5. This settles the tower growth rate for off-diagonal Ramsey numbers for all $r_4(5,n)$6, confirming a conjecture of Erdős and Hajnal [E-H-Con] on their minimum tower type.

Technical Approach and Innovations

The proof proceeds through several stages:

• Probabilistic Coloring Construction: Using a random coloring of pairs in a sufficiently large set, the authors construct a coloring $r_4(5,n)$7 such that for any $r_4(5,n)$8-subset, there is a triple whose pairwise colors satisfy a critical non-symmetry. This is quantified using a partial Steiner $r_4(5,n)$9-system to exploit independence in the random coloring.
• Multi-layer Stepping-up Analysis: The construction leverages stepping-up from graphs to four-uniform hypergraphs, encoding edges via $r_k(k+1,n)$0-sequences determined by the binary representations of vertices. Properties of these sequences—local minima, maxima, monotonicity—enable tight control over the construction, excluding $r_k(k+1,n)$1 cliques and bounding the independence number.
• Structural Extremum Hierarchies: The authors introduce a greedy, layered selection of local maxima within $r_k(k+1,n)$2-sequences, effectively constructing and analyzing multi-layer extremum structures to guarantee the absence of large independent sets and $r_k(k+1,n)$3 subgraphs.

The combination of probabilistic and combinatorial techniques is finely tuned; the dependence $r_k(k+1,n)$4 arises from the recursive depth of the extremum hierarchy and the layered pigeonhole-type arguments. Crucially, this intricate combinatorial structure ensures the edge-defining rules (cases (i)-(iii)) consistently prevent the emergence of forbidden subgraphs while rigorously bounding the independence number.

Numerical Strength and Contradictory Claims

The double-exponential bound $r_k(k+1,n)$5 is markedly stronger than all previous lower bounds for $r_k(k+1,n)$6, which typically involved weaker tower functions or exponential dependencies. The result directly contradicts any conjecture or heuristic that the lower bound could not achieve tower-type growth, decisively confirming the conjectured tower rate.

Furthermore, this work confirms that the classical stepping-up lemma, when sufficiently optimized and combined with multi-layer extremum analysis, is powerful enough to reach the full anticipated lower bound for $r_k(k+1,n)$7, a claim previously unsupported by combinatorial construction.

Implications and Future Directions

Theoretical Implications: The result solidifies the tower-type growth of the off-diagonal hypergraph Ramsey numbers and settles a central question in Ramsey theory. It demonstrates that even for minimal extension beyond the clique size ($r_k(k+1,n)$8), the growth rate is asymptotically maximal, showing a profound combinatorial complexity in these hypergraphs.

Practical Implications: While explicit Ramsey constructions remain largely unfeasible for practical use due to their sheer size, this result provides a rigorous barrier for algorithmic and computational approaches attempting to exploit sparsity or structure in large hypergraphs. Algorithms for searching independent sets or clique avoidance in high-uniformity hypergraphs must account for the immense growth of these extremal functions.

Further Developments: The multi-layer extremum structure introduced in the proof has been successfully applied to the Erdős-Rogers problem [D-H-L-W-1, D-H-L-W-2] and is likely to stimulate additional research. Potential future work includes:

• Refinement of exponent constants (improved dependencies on $r_k(k+1,n)$9).
• Adaptation of techniques to other Ramsey-type extremal functions (e.g., restricted colorings, structural variants).
• Applications to phase transition analysis in random hypergraphs and connectivity thresholds.

Conclusion

This paper resolves the longstanding question on the tower growth rate for off-diagonal hypergraph Ramsey numbers $k\ge 2$0 by establishing a double-exponential lower bound for $k\ge 2$1. The combination of probabilistic coloring, stepping-up techniques, and multi-layer extremum analysis yields a construction that confirms the expected combinatorial complexity of large uniform hypergraphs. The implications reach both the theoretical understanding of Ramsey numbers and the practical limits of hypergraph algorithmics, while the technical innovations promise to impact related extremal problems in combinatorics.