Nearly tight exponents for off-diagonal Ramsey numbers

Abstract: We construct a new family of $K_s$-free graphs that leads to improved lower bounds for Ramsey numbers across a wide range of parameters. For any fixed $s \ge 4$, we show that the off-diagonal Ramsey numbers satisfy $r(s, k) \ge k^{s-2 + o(1)}.$ For $s \ge 6,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen logarithmic improvements. This nearly matches the best known upper bound which is of the form $r(s, k) \le k^{s-1 + o(1)}$ and which is widely believed to give the correct exponent. More generally, we show that if $s, k/s \rightarrow \infty$, then $r(s, k) = \left(\frac{k}{s}\right)^{(1+o(1)) s},$ where the upper follows from the seminal work of Erdős and Szekeres in 1935. We also obtain improved lower bounds for Ramsey numbers extremely close to the diagonal as well as for diagonal multicolor Ramsey numbers.

• The paper introduces a novel construction of Kₛ-free graphs using polarity graphs to nearly match the upper bound exponent for r(s, k).
• It improves classical lower bounds by proving r(s, k) ≥ k^(s−2+o(1)), marking a significant leap from Spencer’s traditional bounds.
• The integration of algebraic and container methods offers practical insights for extremal combinatorics and theoretical computer science.

Nearly Tight Exponents for Off-Diagonal Ramsey Numbers

Overview and Motivation

The paper "Nearly tight exponents for off-diagonal Ramsey numbers" (2605.28793) addresses a long-standing open problem in extremal combinatorics: the asymptotics of the off-diagonal Ramsey numbers $r(s, k)$, where $s$ is fixed and $k \to \infty$. The classical Ramsey number $r(s,k)$ is the minimum $n$ such that every graph on $n$ vertices contains a $K_s$ or an independent set of size $k$. While diagonal Ramsey numbers ($s = k$) have received much attention, the off-diagonal regime ($s$ fixed, $s$0 large) remains less resolved. This paper achieves significant progress by producing improved lower bounds for $s$1 which nearly match known upper bounds, narrowing the gap on the leading exponent that dictates the growth of Ramsey numbers in this regime.

Main Results and Claims

The principal result is that for any fixed $s$2, the off-diagonal Ramsey numbers satisfy

$s$3

improving upon Spencer's classical bound $s$4 for $s$5. This is nearly tight with the best known upper bound $s$6, believed to be correct. The paper further generalizes this, proving that for $s$7, the asymptotics are

$s$8

matching the exponent from the Erdős–Szekeres construction.

The paper also provides improved lower bounds when $s$9 and $k \to \infty$0 are extremely close (near-diagonal regime), as well as for diagonal multicolor Ramsey numbers.

Methodology and Construction

The central advance comes from constructing new families of $k \to \infty$1-free graphs with favorable pseudorandom properties. The approach is inspired by recent algebraic and geometric graph constructions, notably polarity graphs generated from finite projective geometries. The building block is the polarity graph $k \to \infty$2 on $k \to \infty$3, defined so that two vertices are adjacent if their associated vectors are orthogonal under the standard inner product.

The properties of these graphs are pivotal: they have large degree and small second eigenvalue, making them optimal $k \to \infty$4-graphs in terms of pseudo-randomness. While previous works employed the Lovász Local Lemma and random graph processes, these constructions leverage algebraic structure to sidestep natural probabilistic bottlenecks, yielding denser $k \to \infty$5-free graphs.

The paper introduces a forbidden bipartite structure $k \to \infty$6, with the absence of $k \to \infty$7 being key to ensuring $k \to \infty$8-freeness in the product graph construction. The container method is used for counting independent sets, combined with expander mixing arguments on pseudorandom graphs, which enable precise control over the independence number and thus Ramsey thresholds.

Figure 1: The forbidden structure $k \to \infty$9, illustrating green (vertical) edges in $r(s,k)$0 and red (diagonal) edges in $r(s,k)$1, whose absence underpins the $r(s,k)$2-freeness of the constructed graphs.

Strong Numerical Results

For any fixed $r(s,k)$3, the lower bound exponent $r(s,k)$4 contrasts markedly with the previous best of $r(s,k)$5, representing a significant improvement. The construction yields, for sufficiently large $r(s,k)$6, the bound

$r(s,k)$7

showing explicit dependence on $r(s,k)$8 and tightness up to a logarithmic factor. When $r(s,k)$9, the bound matches the upper bound up to $n$0 in the exponent.

For $n$1 near the diagonal ($n$2), the paper gives

$n$3

surpassing Spencer's local lemma approach whenever $n$4.

For the multicolor Ramsey number, an exponential lower bound is achieved:

$n$5

an exponential improvement for any fixed $n$6 over previous constructions.

Theoretical Implications

The results indicate that the correct leading exponent for $n$7 in the off-diagonal regime is likely $n$8, narrowing the prior gap. The new construction paradigm, leveraging algebraic graphs defined over projective spaces and container-based enumeration, sets the stage for further progress toward the conjectured optimal exponent.

The paper also impacts bounds on the Erdős-Rogers function $n$9, demonstrating improved upper bounds when $n$0, using the new Ramsey construction. Furthermore, the techniques offer promising avenues for extending bounds on Ramsey numbers for forbidden subgraphs and multicolor settings.

Practical Implications and Future Directions

Practically, tighter bounds on Ramsey numbers are foundational in the analysis of random graphs, extremal combinatorics, and theoretical computer science, as many algorithmic hardness results depend on such thresholds. Improved lower bounds can inform probabilistic method arguments, derandomization strategies, and explicit constructions in coding theory.

Future developments may center around:

• Closing the remaining gap to the $n$1 exponent for general $n$2 by finding even denser $n$3-free pseudorandom graphs.
• Extending algebraic constructions to multicolor and hypergraph Ramsey regimes.
• Investigating connections between container methods and spectral graph theory to refine independence number estimates.

The paper also suggests exploring possible bipartite analogs, as well as variations forbidding only certain configurations, which could yield further gains in the off-diagonal and near-diagonal regimes.

Conclusion

This work substantially advances the state-of-the-art for off-diagonal Ramsey numbers by providing nearly tight exponents for lower bounds in a regime that had resisted improvement for decades. The synthesis of algebraic constructions with container enumeration methods offers a robust template for future investigations into Ramsey theory and related combinatorial thresholds. The implications span both practical algorithmic contexts and deep theoretical questions in probabilistic combinatorics, indicating rich prospects for additional progress and applications.