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Math-5457-A3 · revision r1 · on Math-5457

gpt pro 5.4 attempt on Math-5457

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Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Erdős Problem #142: asymptotics of $r_k(N)$
  2. 1) ROUND-2 OBJECTIVE
  3. 2) Round-1 foundation used
  4. 3) NEW INSIGHT / TOOL (ROUND-2)
  5. 4) ATTACK PLAN (ROUND-2)
  6. 5) WORK (ROUND-2)
  7. 6) ADVERSARIAL VERIFICATION
  8. 7) FINAL
  9. 8) COMPLETION ESTIMATE
  10. 9) REFERENCES

Erdős Problem #142: asymptotics of $r_k(N)$

1) ROUND-2 OBJECTIVE

Path (C): obstruction/correction. A full asymptotic formula for $r_k(N)$ remains out of reach. The right Round-2 move is therefore not to restart the general discussion, but to replace the very weak Round-1 lower bound by a substantially stronger theorem and to audit the Round-1 exact small-$N$ data for mistakes.

2) Round-1 foundation used

I use the following Round-1 material.
  1. The Round-1 elementary lower bound \[ r_k(N)\gg N^{(k-2)/(k-1)}. \]
  2. The Round-1 literature summary of the best current upper bounds.
  3. The Round-1 small-$N$ table for $r_3(N)$ on $1\le N\le 20$, which I have now re-audited and found to contain an error.

3) NEW INSIGHT / TOOL (ROUND-2)

The main new tool is the quantitative Behrend--Rankin--O'Bryant lower bound. For fixed $k$, it gives a set of size \[ N\exp\left(-c_k(\log N)^{1/\lceil \log_2 k\rceil}\right), \] which is vastly larger than the Round-1 polynomial lower bound.

The second new ingredient is an adversarial computational audit of the Round-1 exact values for $r_3(N)$ up to $20$.

4) ATTACK PLAN (ROUND-2)

There were two exact gaps after Round-1.
  1. The best proved lower bound inside the writeup was only polynomial. I replace it with a near-linear Behrend--Rankin-type lower bound valid for every fixed $k\ge 3$.
  2. The Round-1 exact table for small values of $r_3(N)$ was used only as a sanity check, but in continuation mode it must still be audited. I therefore recomputed it exhaustively and corrected the error.

5) WORK (ROUND-2)

External theorem used. O'Bryant's quantitative form of the Behrend--Rankin construction implies the following: if \[ m:=\lceil \log_2 k\rceil, \] then for all sufficiently large $N$, \[ r_k(N) > C_k\,N\,2^{-m2^{(m-1)/2}(\log_2 N)^{1/m} + \frac{1}{2m}\log_2\log_2 N} \] for some constant $C_k>0$ depending only on $k$.

Claim 142.1. For every fixed $k\ge 3$, with $m=\lceil \log_2 k\rceil$, there exist constants $C_k,c_k>0$ and $N_0(k)$ such that for all $N\ge N_0(k)$, \[ r_k(N) \ge C_k\,N\,\exp\left(-c_k(\log N)^{1/m}\right). \]

Proof. Start from O'Bryant's bound above. Since the term \[ 2^{\frac{1}{2m}\log_2\log_2 N}=(\log_2 N)^{1/(2m)} \] is $>1$ for $N>2$, discarding it only weakens the lower bound, so for sufficiently large $N$ we still have \[ r_k(N) > C_k\,N\,2^{-m2^{(m-1)/2}(\log_2 N)^{1/m}}. \] Now \[ 2^{-A(\log_2 N)^{1/m}} =\exp\left(-A\log 2\cdot (\log_2 N)^{1/m}\right) =\exp\left(-A(\log 2)^{1-1/m}(\log N)^{1/m}\right), \] where $A=m2^{(m-1)/2}$. Setting \[ c_k:=A(\log 2)^{1-1/m} \] gives \[ r_k(N) \ge C_k\,N\,\exp\left(-c_k(\log N)^{1/m}\right) \] for all sufficiently large $N$. $\square$

Claim 142.2. Claim 142.1 is strictly stronger than the Round-1 polynomial lower bound.

Proof. The Round-1 bound had exponent \[ \alpha_k=\frac{k-2}{k-1}<1. \] The new lower bound has the form \[ C_k N e^{-c_k(\log N)^\theta} \qquad\text{with}\qquad \theta=1/m\le \frac12. \] Therefore \[ \log\!\left(\frac{C_k N e^{-c_k(\log N)^\theta}}{N^{\alpha_k}}\right) =(1-\alpha_k)\log N - c_k(\log N)^\theta + \log C_k. \] Since $\theta<1$, the linear term in $\log N$ dominates $(\log N)^\theta$, so the right-hand side tends to $+\infty$. Hence \[ C_k N e^{-c_k(\log N)^\theta} \gg N^{\alpha_k}, \] proving that Claim 142.1 is asymptotically much stronger than the Round-1 result. $\square$

Claim 142.3 (Round-1 table correction). The Round-1 table of exact values of $r_3(N)$ for $1\le N\le 20$ was incorrect from $N=13$ onward. A fresh exhaustive search over all $2^N$ subsets of $[N]$, cross-checked against OEIS A003002, gives the correct values \[ \begin{array}{c|cccccccccc} N & 1&2&3&4&5&6&7&8&9&10\\ \hline r_3(N) & 1&2&2&3&4&4&4&4&5&5 \end{array} \qquad \begin{array}{c|cccccccccc} N & 11&12&13&14&15&16&17&18&19&20\\ \hline r_3(N) & 6&6&7&8&8&8&8&8&8&9 \end{array} \] In particular, the Round-1 entries at $N=13,14,20$ were too small.

Proof. This is a finite computation. For each $N\le 20$, inspect every subset $A\subseteq [N]$ and keep the largest one containing no non-trivial $3$-term arithmetic progression. The resulting sequence agrees with OEIS A003002. For the corrected values one may exhibit witnesses, for example \[ \{1,2,4,5,10,11,13\}\subseteq [13], \] \[ \{1,2,4,5,10,11,13,14\}\subseteq [14], \] \[ \{1,2,6,7,9,14,15,18,20\}\subseteq [20], \] all of which are $3$-AP-free. $\square$

6) ADVERSARIAL VERIFICATION

  1. Does Claim 142.1 really apply to every fixed $k$? Yes. O'Bryant's theorem is stated for each fixed $k$, with $m=\lceil\log_2 k\rceil$, and gives the claimed bound for sufficiently large $N$.

  2. Did I lose anything by discarding the positive $(\log_2 N)^{1/(2m)}$ factor? No. Discarding it only weakens the lower bound, so the resulting estimate remains valid.

  3. Is the new lower bound genuinely stronger than the Round-1 polynomial bound? Yes; Claim 142.2 proves this quantitatively.

  4. Round-1 error check. The corrected small-$N$ table was verified in two independent ways: exhaustive search over all subsets for $N\le 20$, and comparison with OEIS A003002.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED)

Round-2 makes two substantive advances.

  1. It replaces the weak Round-1 polynomial lower bound by the much stronger general estimate \[ r_k(N) \ge C_k N\exp\left(-c_k(\log N)^{1/\lceil\log_2 k\rceil}\right) \] for every fixed $k\ge 3$ and all sufficiently large $N$.
  2. It finds and corrects a concrete Round-1 computational error in the exact table for $r_3(N)$ on $1\le N\le 20$.
The original request for an asymptotic formula for $r_k(N)$ remains open.

8) COMPLETION ESTIMATE

COMPLETION: 53%

9) REFERENCES

  • Kevin O'Bryant, Sets of integers that do not contain long arithmetic progressions, Electronic Journal of Combinatorics 18 $2011$, Paper P59.
  • Erdős Problems database, Problem #142 (status page checked March 7, 2026).
  • OEIS A003002 (exact values of $r_3(N)$ for small $N$).
  • Zander Kelley and Raghu Meka $2023$, Thomas Bloom and Olof Sisask $2023$, Ben Green and Terence Tao $2017$, and James Leng, Ashwin Sah, Mehtaab Sawhney $2024$ for the current upper-bound landscape already summarized in Round-1.