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Statement

With $g(b,r)$ the least $a$ such that $a\to(b,b)^r$, and $a*b=a^b$ evaluated from the right (printed p. 139), the paper states after 16.4:

"It is reasonable to conjecture that, in fact,

$$g(b,3)\ge2^{2^{c_3b}}$$

for some absolute real constant $c_3>0$ and that, more generally,

$$(1)\qquad g(b,r)\ge2*2**2*(c_rb)\qquad(r\ \text{"factors"})$$

for some real positive $c_r$ which is independent of $b$."

The first display is the statement of Problem 564 with $R_3(b)=g(b,3)$. The page continues: "By means of the methods of section 14 we can prove the following 'stepping-up' lemma. Lemma 6. There is a real number $c\ge1/10$ such that $g(b,r)\ge2*(cg(b,r-1))$ for $r>4$." Then: "Using this lemma we can deduce from 16.2 that for $r\ge3$

$$g(b,r)\ge2*2**2*(\tfrac12c^{r-3}b)\qquad(r-1\ \text{"factors"}).$$

This result approaches the conjecture (1) but a big gap still exists in the case $r=3$ between the conjecture and the established estimate. Since these results are obviously not final we omit the proofs."

The range of Lemma 6 is printed "for $r>4$" (a plain "$>$", unlike the "$\ge$" glyphs beside it, checked on a 300 dpi crop). The deduction that follows starts from 16.2 ($r=2$) and needs the lemma from $r=4$ on to reach the displayed tower of height $r-1$, so the intended range appears to be $r\ge4$; the page records the text as printed. Stepping up from $r-1=2$ to $r=3$ is not covered by the lemma, which is why the $r=3$ case remains at the single-exponential 16.4.

Source. P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93--196; printed p. 140 (PDF p. 48 of the scan), read on the page image with 300 dpi crops. The scan's text layer is garbled.

Read depth. Claims checked: the conjecture, display (1), Lemma 6, the deduced bound and the closing remark were read clause by clause on the page image. The paper omits every proof of the section; nothing is proof-checked.

Proof pointer

None: a conjecture, and a lemma whose proof the paper omits ("by means of the methods of section 14").

Dependencies

Lemma 6 and the deduced bound rest on 16.2 ($g(b,2)\ge2^{b/2}$, Erdős 1947, the paper's [9]) and on the omitted stepping-up argument.

Bears on

• Problem 564: the problem's own statement in its 1965 wording, together with the authors' record that the $r=3$ case is where the gap between conjecture and estimate remains.
• Problem 562: the general conjecture (1), a tower of height $r$ with top $c_rb$ as a lower bound for $g(b,r)$, is the problem's statement in its 1965 wording for every $r\ge3$; the deduced bound below it is a tower of height $r-1$, whose $(r-1)$-fold iterated logarithm is $\log_2(\tfrac12c^{r-3}b)$, of order $\log b$, so for every $r\ge3$ the 1965 estimate leaves the problem's lower side at order $\log n$ against the conjectured order $n$.