GPT 6 Astra Ultra attempt on Math-5478
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Contents
Erdős Problem 179: many short progressions without a long one
1) Statement and conventions
For fixed $k<\ell$, let $F_k(N,\ell)$ be the least integer threshold such that every $N$-element set of positive integers with at least that many $k$-term arithmetic progressions contains an $\ell$-term progression. We count nonconstant progressions once, in increasing order. The main questions are whether $F_3(N,4)=o(N^2)$ and whether $\log F_3(N,\ell)/\log N\to2$ for each fixed $\ell>3$. Both answers are affirmative. This note deduces them from the precise Fox--Pohoata comparison theorem and standard progression bounds.2) The threshold differs from a maximum by one
Let $M_{k,\ell}(N)$ be the largest number of $k$-term progressions in an $\ell$-progression-free set of $N$ integers. Such sets exist for every $N$ when $\ell\geq3$: distinct powers of $3$ contain no three-term progression. Indeed, $3^a+3^c=2\cdot3^b$ with $a<b<c$ is impossible after dividing by $3^a$ and reducing modulo $3$.For $k\geq2$, each progression is determined by its first two terms, so its count is at most $\binom N2$. Thus the nonempty set of attainable counts is a finite set of integers and has a maximum, even though the ambient integers are unbounded. The definition of the threshold gives \[ F_k(N,\ell)=M_{k,\ell}(N)+1. \tag{1} \] A translation shows that allowing arbitrary integers instead of positive integers makes no difference.
3) External inputs and the asymptotic deduction
Let $r_\ell(N)$ be the largest size of an $\ell$-progression-free subset of $[N]$. Fox--Pohoata's Theorem 1.2, with constants enlarged if the progression-count convention differs, gives for fixed $\ell>k\geq3$ \[ c_k\left(\frac{r_\ell(N)}N\right)^{2(k-2)}N^2 \leq M_{k,\ell}(N) \leq C_k\left(\frac{r_\ell(N)}N\right)^\gamma N^2, \tag{2} \] for sufficiently large $N$, where $\gamma>0$. We use $2$ as an external theorem. The other external inputs are Szemerédi's theorem $r_\ell(N)=o(N)$ and Behrend's lower bound $r_3(N)\geq N\exp(-C\sqrt{\log N})$ for large $N$.An $\ell$-term progression contains a three-term progression, so $r_\ell(N)\geq r_3(N)$ for $\ell\geq3$. Substitution in the lower bound of $2$ gives \[ M_{k,\ell}(N)\geq N^2\exp(-C_{k,\ell}\sqrt{\log N}). \] Its upper bound, together with Szemerédi, gives $M_{k,\ell}(N)=o(N^2)$. Consequently $1$ implies both \[ F_k(N,\ell)=o(N^2),\qquad 2-\frac{C_{k,\ell}}{\sqrt{\log N}} \leq\frac{\log F_k(N,\ell)}{\log N}\leq2+o(1). \] In particular the two explicitly asked assertions hold. There is no conflict between an exponent tending to two and density tending to zero relative to $N^2$.
4) The degenerate short-length cases
The restriction $k\geq3$ in $2$ matters. Under the singleton convention for one-term progressions, every $N$-set has exactly $N$ of them, so $F_1(N,\ell)=N+1$ for $\ell\geq3$. Every pair is a two-term progression, hence $F_2(N,\ell)=\binom N2+1$ for $\ell\geq3$. Neither should be assigned the preceding little-$o$ conclusion. When $\ell=2$, every $N$-set with $N\geq2$ already contains the desired pair, independently of any counting threshold.5) Outcome and sources
Outcome: SOLVED using the three stated external inputs. The threshold convention, range of parameters, and limit deductions are proved explicitly. The additive-combinatorial proof of $2$ and the two classical progression theorems are not reproved here.- J. Fox and C. Pohoata, Sets without $k$-term progressions can have many shorter progressions, Theorem 1.2,
https://arxiv.org/abs/1908.09905; its references include the classical bounds used above. - T. F. Bloom, Problem 179,
https://www.erdosproblems.com/179, checked 23 September 2026.