Erdos 1997 some my favorite problems results

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conjecture_p54: Erdős's 1997 statement of his conjecture that every finite Sidon sequence extends to a Singer perfect difference set modulo p^2+p+1 for some prime power p, his remark that it is perhaps too optimistic, and the weaker conjecture on completing to a Sidon sequence with a_n < (1+ε) n^2; the origin wording of Problem 707.

conjecture_p65: Erdős's 1997 statement that for n points forming a convex polygon, with s_i the number of pairs at the i-th distance, he conjectured and Fishburn proved that the sum of the squares of the s_i is below c n^3, with the regular n-gon conjecture and the prize conjecture (5.3) for point sets without convexity; the origin wording of Problem 94.

display_2_20: Erdős's 1997 dilation question: a sequence of reals above one with |kα_i − α_j| ≥ 1 for all i ≠ j and all k generalizes a primitive sequence; the printed question on the reciprocal sums, whose signs are the reverse of the site's, and the prize offer; the origin wording of Problem 143.

display_2_21: Erdős's 1997 prime-gap passage: his 1934 bound, Rankin's 1938 factor, the conjecture that the Rankin bound holds for every constant c infinitely often (the offer lowered) and the new offer for d_n > (log n)^{1+ε}; the origin wording of Problem 4.

display_4_2: Erdős's 1997 statement of the Erdős–Szekeres bounds on the diagonal Ramsey number, his offers of prizes for the existence and for the value of the limit of f(n)^{1/n} with the guess c = 2, and his offer for a constructive exponential lower bound; the origin wording of Problems 77 and 78.

display_4_3: Erdős's 1997 statement of the Erdős–Hajnal–Rado bounds on the two-color Ramsey number of the complete 3-uniform hypergraph, his belief that the double-exponential upper bound is the truth, the Erdős–Hajnal unbalanced triples result that seems to favor the lower bound, the criterion that would make him doubt the upper bound, and Hajnal's four-color bound r_3(n,n,n,n) > 2^{c 2^n}; the site's form of the bounds for Problem 564.

display_4_4: Erdős's 1997 statement of the Erdős–Graham question whether every k-coloring of the integers at least 2 has a finite monochromatic set with reciprocal sum one, with the threshold f(k), the every-positive-rational conjecture, and the density forms: a reciprocal sum growing faster than (log log n)^2 should force a unit subsum, and a reciprocal sum above c log n is believed to; the origin wording of Problems 46 and 47.

display_4_5: Erdős's 1997 statement, attributed to Simonovits and himself, of the degenerate Turán conjecture with the parameter shifted by one from the site's, the remark that it is open even for r = 3, the companion lower-bound conjecture for graphs with an induced subgraph of minimum degree at least r, and the prize offers; the origin wording of Problem 146.

problem_p51: Erdős's 1997 statement of the progression-free function r_k(n), the bounds on r_3(n) he lists as current records, and his offers of prizes for r_3(n) < n/(log n)^c for every c and for any asymptotic formula for r_k(n); the origin wording of Problems 140 and 142.

problem_p62: Erdős's 1997 account of the off-diagonal Ramsey numbers: the bounds c_1 n^2/log n < r_2(3,n) < c_2 n^2/log n credited to Ajtai, Komlós and Szemerédi and to Kim, the wish for an asymptotic formula for r_2(3,n), his former expectation r_2(4,n) > n^{3−ε} and r_2(k,n) > n^{k−1}/(log n)^2 now doubted, Spencer's c n^{5/2} and the printed upper bound r_2(ℓ,n) < c n^{ℓ−1}/log n; the origin wording of Problems 165 and 166.

remark_p58: Erdős's 1997 sentence that he and Ricci proved the set of limit points of the normalized prime gaps d_n/log n has positive measure, with the expectation that the limit points are everywhere dense and the note on Maier's partial results; the origin wording of Problem 5.

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Paul Erdős, Some of My Favorite Problems and Results. In: R. L. Graham and J. Nešetřil (eds.), The Mathematics of Paul Erdős I, Algorithms and Combinatorics 13, Springer, Berlin Heidelberg (1997), 47--67, chapter DOI 10.1007/978-3-642-60408-9_3; the author at the Mathematical Institute, Hungarian Academy of Sciences, Budapest (head of p. 47). The chapter opens Part I, "Early Days", of the volume (contents, p. XI), after the editors' introduction (pp. 45--46), which presents it as a typical Erdős problem article of the period, unusual only in ranging over all of his fields (p. 46). Cited as [Er97c] on the problem pages, the site's key. The volume's copyright page prints "© Springer-Verlag Berlin Heidelberg 1997", "Softcover reprint of the hardcover 1st edition 1997", ISBN-13 978-3-642-64394-1, e-ISBN-13 978-3-642-60408-9 and the volume DOI 10.1007/978-3-642-60408-9; the editors' memorial note (an unnumbered page, PDF p. 6, between the copyright page and the Preface on p. V) records that, the week before the volumes were scheduled to go to press, they learned that Erdős had died on 20 September 1996. The chapter's 21 references (pp. 66--67) are books, survey chapters, Erdős's own papers and four journal papers by others (Choi [2], Crocker [3], Gallagher [14] and Romanoff [20]); among them [12] is erdos_1980_old_new_problems_results_combinatorial_number_theory and [19] is the volume from which erdos_1990_problems_results_graphs_hypergraphs_similarities_differences is extracted.

The copy read for this card is the whole eBook of Volume I, from which the chapter was read: 413 PDF pages (front matter pp. I--XVI and printed pp. 1--399, with unnumbered leaves; the last article ends on printed p. 399 = PDF p. 410, and PDF pp. 411--413 are two unnumbered series-list pages and the publisher's closing page, checked on the page images), the publisher's scan of the printed volume (the eBook's metadata names an Acrobat 9.0 Paper Capture plug-in and an August 2011 creation date) with an OCR text layer that reads the prose and garbles displays, diacritics and some names ("Thran" for Turán, "2../2" for $2^{\sqrt2}$). The chapter is printed pp. 47--67 = PDF pp. 62--82 (printed p. $n$ is PDF p. $n+15$); p. 47 is the chapter's title page and p. 67 ends the reference list. Provenance: the whole eBook was obtained from the publisher on 2026-09-22 as a DRM-free PDF through the library's acquisition, from https://doi.org/10.1007/978-3-642-60408-9_3 (the chapter's DOI, resolving to the publisher's platform); 41,797,026 bytes. The eBook, the whole volume, prints "© Springer-Verlag Berlin Heidelberg 1997" and a notice that the work is subject to copyright with "All rights are reserved, whether the whole or part of the material is concerned", naming translation, reprinting and reuse of illustrations among them, on the volume's copyright page (PDF p. 5), which covers the chapter, every other right reserved.

Read status: claims checked for the passages the citing problems consume, each read clause by clause on the page images of PDF pp. 62, 65--66, 69, 72--73 and 77--80 (printed pp. 47, 50--51, 54, 57--58 and 62--65) on 2026-09-22: the title page (p. 47), the definition of $r_k(n)$ (p. 50), the $r_3(n)$ bounds and the two offers (p. 51), the Singer passage and the Sidon completion conjecture (p. 54), display (2.20) and its question (p. 57) with the offer (p. 58), the prime-gap passage with (2.21) and the Ricci sentence (p. 58), the Ramsey notation, (4.2), the limit offers, the constructive offer and the $r_2(3,n)$ bounds (p. 62), the $r_2(4,n)$ and $r_2(\ell,n)$ bounds, (4.3) with its discussion and (4.4) with the reciprocal-sum questions (pp. 63--64), (4.5) and (4.6) (p. 64) and the Geometry section through (5.3) (p. 65). The rest of the chapter (pp. 48--49, 52--53, 55--56, 59--61 and 66--67) was read in the text layer for its topics, offers and reference list, and the volume's title, copyright and contents pages in the text layer for identification. The chapter is a problem survey and proves nothing: every result it reports (its own and others') is stated without proof, and nothing here is independently reviewed.

Contents

The chapter has five sections: 1. Introduction (p. 47); 2. Number theory (pp. 47--59); 3. Polynomials (pp. 59--60); 4. Combinatorics (pp. 60--64); 5. Geometry (pp. 65--66); references (pp. 66--67). The introduction tells the Hilbert anecdote on $2^{\sqrt2}$ and notes that many of the problems have come to carry cash prizes.

• § 2, pp. 47--49 (text layer): the distinct-subset-sums function $f(n)$ with the conjecture (2.1) $f(n)<\log n/\log2+c$, the Erdős--Moser bound and the Conway--Guy example; covering systems (2.3) with the question whether the least modulus can be arbitrarily large, Choi's system with $n_1=20$, the odd-moduli question and Schinzel's question for a covering system with $n_i\nmid n_j$ for $i\ne j$, and the $2^k+p$ history (Romanoff, Crocker, Gallagher); the Erdős--Selfridge theorem on products of consecutive integers.
• § 2, pp. 50--52 (pp. 50--51 page images): the Tauberian theorem (2.4); van der Waerden numbers $W(k)$, Shelah's primitive recursive bound, $W(k)>2^{k/2}$ (Erdős--Rado), Berlekamp's $W(p+1)\ge p\cdot2^p$, Graham's tower offer. Then, quoted (p. 50): "let $r_k(n)$ be the smallest integer for which every sequence of integers $1\le a_1<a_2<\cdots<a_t\le n$ with $t\ge r_k(n)$ contains an arithmetic progression of $k$ terms"; the Erdős--Turán conjecture $r_k(n)/n\to0$; p. 51: Salem--Spencer $r_3(n)>n^{1-c\log\log n}$ (so printed, for $n^{1-c/\log\log n}$), Behrend $r_3(n)>n\exp(-c\sqrt{\log n})$ ("the current record"), Roth $r_3(m)\le cn/\log\log n$ (so printed), Heath-Brown and Szemerédi $r_3(n)\le n/(\log n)^\alpha$, $\alpha\approx 1/4$; the two offers paged on problem_p51 (a prize for $r_3(n)<n/(\log n)^c$ for every $c$, a prize for an asymptotic formula for $r_k(n)$, which he calls probably unattackable at present) and quoted under #142 below. Szemerédi's 1974 theorem and its prize, Furstenberg; the conjecture (2.5) that $\sum1/a_k=\infty$ forces arbitrarily long progressions; consecutive primes in progression; p. 52: display (2.6) on $\sum1/a_i$ against $\ln W(k)$ for $k$-progression-free sequences, and $W(3)=9$, $W(4)=35$, $W(5)=178$.
• § 2, pp. 52--55 (p. 54 page image): Sidon's two questions; $c_1\log n<f(n)<c_2\log n$ (2.7) by a random sequence, the Erdős--Turán conjecture $\limsup f(n)=\infty$, a prize for a constructive answer to Sidon's first question, (2.8) with a prize, the Erdős--Sárközy result (2.9); Sidon sequences, the greedy bound (2.10), the Ajtai--Komlós--Szemerédi bound (2.11), the Erdős--Rényi construction; the triple-sums conjecture; $S(n)<n^{1/2}+cn^{1/4}$ (2.12), $S(n)>n^{1/2}-n^{1/2-\epsilon}$ (2.13) from Singer, the conjectures (2.14) and (2.15) with a prize for "clearing up these two problems"; Singer's theorem as Erdős states it (p. 54): for a prime power $p$ there are $p+1$ residues $a_1,\ldots,a_{p+1}\pmod{p^2+p+1}$ whose differences $a_i-a_j$, $1\le i,j\le p+1$, represent every nonzero residue $t\pmod{p^2+p+1}$ exactly once, which he says easily gives (2.13); the Erdős--Fuchs theorem; the Sidon completion conjecture, paged on conjecture_p54, with its weaker form and (2.16); p. 55: $\limsup a_n/(n^2\log n)>0$ (2.17), the conjecture (2.18), the Erdős--Sárközy conjecture (2.19), and the $h(n)$ question.
• § 2, pp. 55--58 (pp. 57--58 page images): the $\pm1$ discrepancy conjecture (for every $c$ a $d$ with $\max_n|\sum_{k=1}^nf(kd)|>c$) and its stronger form, printed $\max_n|\sum_{k=1}^{\ell}f(kd)|>c\log n$ with $\ell<n/d$ under the sum; primitive abundant numbers, the Hardy--Ramanujan theorem, the three-series theorem and the Erdős--Kac story (pp. 56--57); Davenport--Erdős on sequences of multiples and the primitive-sequence bound; display (2.20) and its question, paged on display_2_20 (p. 58); divisors in $(n,n^{1+\epsilon_n})$ and the Maier--Tenenbaum theorem; the primes: prime gaps and (2.21), paged on display_2_21; the Ricci sentence on limit points of $d_n/\log n$, paged on remark_p58; the Erdős--Kalmár elementary bounds (2.22)--(2.23) and the Rosser story (pp. 58--59).
• § 3, pp. 59--60 (text layer): the Erdős--Gallai integral inequality, the Erdős--Herzog--Piranian polynomial with small $|f_n(z)|<1$ set, the Erdős--Offord count of real roots, the Erdős--Clarkson theorem on Müntz approximation, and the derivative bound $en/2$ for polynomials with real roots outside $(-1,1)$.
• § 4, pp. 60--61 (text layer): the Erdős--Faber--Lovász conjecture with Kahn's $(1+o(1))n$; strong and weak $\Delta$-systems, $2^n<f(n,3)< 2^nn!$, the conjecture (4.1) $f(n,3)<c_3^n$, Kostochka's bound and the Axenovich--Fon-der-Flaass--Kostochka bound $f(n,3)<(n!)^{1/2+ \epsilon}$; the infinite-chromatic-number graph with $f(n)$ edges to bipartiteness; cycle lengths in a density-zero set $A$.
• § 4, pp. 62--63 (page images), Ramsey's theorem: $r_k^{(\ell)} (p_1,\ldots,p_\ell)$, Rado's arrow notation and the square-bracket notation of Erdős, Hajnal and Rado; the Erdős--Szekeres bounds (4.2), $cn2^{n/2}<r_2(n,n)<\binom{2n-2}{n-1}$; $f(n)=r_2^2(n,n)$ with the offers for $\lim f(n)^{1/n}$, Spencer's and Thomason's improvements and the constructive offer, paged on display_4_2; the bounds $c_1n^2/\log n<r_2(3,n)<c_2n^2/\log n$ (Ajtai--Komlós--Szemerédi and Kim), the wish for an asymptotic formula, the retracted expectation $r_2(4,n)>n^{3-\epsilon}$ and $r_2(k,n)>n^{k-1}/(\log n)^2$, Spencer's $cn^{5/2}$ and the printed $r_2(\ell,n)<cn^{\ell-1}/\log n$, paged on problem_p62; the Erdős--Hajnal--Rado bounds (4.3), $2^{cn^2}<r_3(n,n)<2^{2^n}$, with the unbalanced-triples result, the doubt criterion and Hajnal's four-color bound, paged on display_4_3.
• § 4, pp. 63--64 (page images): the Erdős--Graham unit-fraction question (4.4) with $f(k)$, the every-positive-rational conjecture and the reciprocal-sum thresholds $(\log\log n)^2$ and $c\log n$, paged on display_4_4; extremal graph theory: the Turán number $T_n(H)$, the degenerate conjecture (4.5) with its companion and the offers, paged on display_4_5; $f_n(H)$ with the conjecture (4.6) $f_n(H)<2^{(1+o(1))T_n(H)}$, open for $C_4$, Kleitman--Winston's $f_n(C_4)<2^{cn^{3/2}}$, and the two-copies conjecture.
• § 5, pp. 65--66 (p. 65 page image): distinct distances (5.1) $f(n)> cn/\sqrt{\log n}$ and unit distances (5.2) $g(n)<n^{1+c/\log\log n}$, with "The best results so far are $f(n)>n^{3/4}$ and $g(n)<n^{5/4}+\epsilon$ [sic] for any $\epsilon>0$ and $n>n_0(\epsilon)$"; the equidistant-points function $h(n)=o(n^\epsilon)$? (a prize for a proof, a prize for a counterexample); Szemerédi's $\lfloor n/2\rfloor$ conjecture; the convex polygon distance sum with Fishburn's theorem and (5.3), paged on conjecture_p65; p. 66 (text layer): lines through four points with no five collinear the Erdős--Purdy $h(n)$ question, and the Erdős--Klein--Szekeres problem with Szekeres's conjecture $f(n)=2^{n-2}+1$.

Compiled scope

The chapter is compiled at statement depth for the passages the fifteen citing problem pages consume, read clause by clause on the page images and quoted or restated above or on the eleven result pages. The other passages are summarized from the text layer for identification only. The chapter proves nothing, and nothing here is independently reviewed. The prizes are recorded as printed in 1997; several differ from the site's (the site's prize for Problem 4 is here smaller, moved to (2.21); Problem 142's asymptotic formula carries a smaller prize here; Problems 46, 94 and 707 carry no offer here).

Bears on. #4, as the site's source Er97c: p. 58 (PDF p. 73), "I used to offer $10000 for a proof that $d_n>c\log n\log\log n\log\log\log\log n/(\log\log\log n)^2$ holds for every $c$ and infinitely many $n$. However, I would now like to offer only $5000 for this conjecture, and instead offer $10000 for a proof that $d_n>(\log n)^{1+\epsilon}$ (2.21) for some $\epsilon>0$", with Erdős's 1934 bound and Rankin's 1938 factor; paged on display_2_21. #5, as the site's source Er97c: p. 58 (PDF p. 73), "Ricci and I proved that the set of limit points of $d_n/\log_n$ [sic] has positive measure. No doubt they are everywhere dense" ($\log_n$ so printed, for $\log n$), with the note that the strongest partial results are Maier's and that Erdős has paid him a prize on more than one occasion for some of them; paged on remark_p58. #46, as the site's source Er97c: p. 63 (PDF p. 78), introduced as an old conjecture of Graham and Erdős that he places where Ramsey theory meets number theory, and posed as: "Is it true that no matter how one $k$-colors the integers $\ge2$ one can always find a solution to $1=\sum_{a\in A}\frac1a$, for a finite monochromatic subset $A$? (4.4)", which they could not prove even for $k=2$, with the threshold $f(k)$ and the every-positive-rational conjecture; no prize is printed for (4.4); paged on display_4_4. #47, as the site's source Er97c: pp. 63--64 (PDF pp. 78--79), the reciprocal-sum thresholds after (4.4): for integers $1<a_1<\cdots\le a_k\le n$ with $\sum_{i\le k}1/a_i>f(n)$, whether $f(n)/(\log\log n)^2\to\infty$ forces a subsequence of the $a_i$ whose reciprocals sum to exactly $1$; the "strongest conjecture", an absolute constant $c$ with the threshold $(c\pm\epsilon)(\log\log n)^2$; and the belief that $\sum_{a_i<n}\frac1{a_i}>c\log n$ already forces a solution of (4.4) among the $a_i$, the problem's $\delta\log N$ form; paged on display_4_4. #77, as the site's source Er97c: p. 62 (PDF p. 77), with $f(n)$ the smallest integer satisfying $f(n)\to(n)_2^2$, that is $f(n)=r_2^2(n,n)$: "I offer $100 for a proof that $\lim_{n\to\infty}f(n)^{1/n}$ exists, and $250 for the value $c$ of this limit"; (4.2) gives $\sqrt2\le c\le4$, he asks "Perhaps $c=2$?", and he remarks that these questions had seen almost no progress; paged on display_4_2. #78, as the site's source Er97c: p. 62 (PDF p. 77), noting that his proof of the lower bound in (4.2) is nonconstructive: "I offer $100 for a constructive proof that $f(n)>(1+\epsilon)^n$", with Frankl and Wilson's constructive $f(n)>n^{c\log n}$ as the record; paged on display_4_2. #94, as the site's source Er97c: p. 65 (PDF p. 80): for $x_1,\ldots,x_n$ the vertices of a convex polygon in the plane, with $s_i$ the number of the $\binom n2$ distances $d(x_i,x_j)$ equal to the $i$-th distance $u_i$ (so $\sum_is_i=\binom n2$), "I conjectured and Fishburn proved that $\sum_is_i^2<cn^3$", followed by the regular $n$-gon conjecture and (5.3) for point sets not assumed convex; the theorem is stated without proof or reference, Fishburn's paper "Distances in Convex Polygons" being in Volume II of the same work (contents, p. XV); paged on conjecture_p65. #140, as the site's source Er97c: p. 51 (PDF p. 66), "I offer $500 for a proof that $r_3(n)<n/(\log n)^c$ for every $c$", the problem's statement and its prize, in the chapter's $r_3(n)$ (the smallest size forcing a 3-term progression, one more than the site's $r_3(N)$, which leaves the bound unchanged); paged on problem_p51. #142, as the site's source Er97c: pp. 50--51 (PDF pp. 65--66), the definition of $r_k(n)$ as the smallest $t$ forcing a $k$-term progression (one more than the site's largest progression-free size), the $r_3(n)$ bounds, and "I offer $500 for a proof that $r_3(n)<n/(\log n)^c$ for every $c$, and $1000 for any asymptotic formula for $r_k(n)$. This is probably unattackable at present"; paged on problem_p51. #143, as the site's source Er97c: pp. 57--58 (PDF pp. 72--73): for real numbers $1<\alpha_1<\alpha_2<\cdots$ with $|k\alpha_i-\alpha_j|\ge1$ for all $i\ne j$ and all $k$ (display (2.20)), a condition which for integers $\alpha_i$ says that no $\alpha_i$ divides another $\alpha_j$, the question as posed: "Does (2.20) imply $\sum_i\frac1{\alpha_i\log\alpha_i}=\infty$ or $\frac1{\log x}\sum_{\alpha_i<x}\frac1{\alpha_i}\to\infty$ as $x\to\infty$? [sic]" and "I offer $500 for settling this annoying diophantine problem"; the printed question has the signs reversed relative to the site's statement (recorded on the result page); paged on display_2_20. #146, as the site's source Er97c: p. 64 (PDF p. 79), "Simonovits and I conjectured long ago that if $H$ is bipartite and every induced subgraph of $H$ has a vertex of degree $<r$, then $T_n(H)<cn^{2-1/(r-1)}$. (4.5)", which he says is open even for $r=3$, with the companion lower-bound conjecture and "I offer $500 for a proof or disproof of each of our conjectures"; the printed parameter $r$ is the site's $r$ plus one, so (4.5) is the site's statement and "open even for $r=3$" is its case $r=2$; the attribution to Simonovits and Erdős jointly is in Erdős's own words; paged on display_4_5. #165, as the site's source Er97c: p. 62 (PDF p. 77), the bounds $\frac{c_1n^2}{\log n}<r_2(3,n)<\frac{c_2n^2}{\log n}$ stated as now known, the upper bound credited to Ajtai, Komlós and Szemerédi and the recent lower bound to J. H. Kim's probability arguments, then the wish as posed: "It would be nice to have an asymptotic formula for $r_2(3,n)$"; no prize is attached to this wish here; paged on problem_p62. #166, as the site's source Er97c: pp. 62--63 (PDF pp. 77--78), his former expectation that probabilistic arguments would establish $r_2(4,n)>n^{3-\epsilon}$ and, for each fixed $k$ and all large $n$, $r_2(k,n)>\frac{n^{k-1}}{(\log n)^2}$, retracted in his words: "It now seems that I am wrong and new ideas will be required", with Spencer's $cn^{5/2}$ as the current lower-bound record for $r_2(4,n)$ and the upper bound $r_2(\ell,n)<cn^{\ell-1}/\log n$ for fixed $\ell$ credited to the Ajtai--Komlós--Szemerédi proof; the chapter states the problem as a former expectation, not as a conjecture with a prize, and its Spencer bound carries no logarithmic factor; paged on problem_p62. #564, as the site's source Er97c: p. 63 (PDF p. 78), the Erdős--Hajnal--Rado bounds $2^{cn^2}<r_3(n,n)<2^{2^n}$ (display (4.3)), then "We believe the upper bound is closer to the truth", qualified by an Erdős--Hajnal result that seems to favor the lower bound, the unbalanced triples result with $s=[(\log n)^{1/2}]$, the doubt criterion, and the four-color bound $r_3(n,n,n,n)>2^{c2^n}$, credited to Hajnal and called very strong evidence for the upper bound in (4.3); the site's form of the 1965 bounds, $2^{cn^2}<R_3(n)<2^{2^n}$, is (4.3) as printed here, and the four-color bound the site attributes to the 1984 book is attributed here to Hajnal, without proof or reference; paged on display_4_3. #707, as the site's source Er97c: p. 54 (PDF p. 69), the conjecture Erdős says he made many years ago: for every finite Sidon sequence $a_1<\cdots<a_t$ there are a prime power $p=q^\alpha$ and an extension $a_1<\cdots<a_t<a_{t+1}<\cdots<a_{p+1}<p^2+p+1$ forming a Singer perfect difference set modulo $p^2+p+1$; he adds that he now feels the conjecture "is perhaps too optimistic", followed by the weaker $(1+\epsilon)n^2$ conjecture and (2.16); the modulus here allows a prime power $p=q^\alpha$ where the site's statement says a prime, and no prize is printed for it; paged on conjecture_p54.

Results. All are statements without proof.

• Problem (p. 51): the offer for $r_3(n)<n/(\log n)^c$ for every $c$ and the offer for an asymptotic formula for $r_k(n)$.
• Conjecture (p. 54): every finite Sidon sequence completes to a Singer perfect difference set modulo $p^2+p+1$, $p=q^\alpha$; its weaker form and (2.16).
• Display (2.20): the dilation condition $|k\alpha_i-\alpha_j|\ge1$ and the reciprocal-sum question, with a prize.
• Display (2.21): $d_n>(\log n)^{1+\epsilon}$ infinitely often, with a prize, and the Erdős--Rankin conjecture with a smaller one.
• Remark (p. 58): the Erdős--Ricci positive-measure theorem for the limit points of $d_n/\log n$ and the expectation that they are dense.
• Display (4.2): $cn2^{n/2}<r_2(n,n)<\binom{2n-2}{n-1}$, the offers for $\lim f(n)^{1/n}$ and for a constructive $f(n)>(1+\epsilon)^n$.
• Problem (pp. 62--63): the $r_2(3,n)$ bounds with the asymptotic-formula wish, and the retracted expectation on $r_2(4,n)$ and $r_2(k,n)$.
• Display (4.3): $2^{cn^2}<r_3(n,n)<2^{2^n}$ and the evidence on each side.
• Display (4.4): the monochromatic unit-fraction question and the reciprocal-sum thresholds.
• Display (4.5): the degenerate Turán conjecture $T_n(H)<cn^{2-1/(r-1)}$ and its companion, with a prize for each.
• Conjecture (p. 65): $\sum s_i^2<cn^3$ for convex polygons, stated as proved by Fishburn, with the regular $n$-gon conjecture and (5.3).

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