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declared openerdos:131
Let F(N)F(N) be the maximal size of A⊆{1,…,N}A\subseteq\{1,\ldots,N\} such that no a∈Aa\in A divides the sum of any nonempty subset of A∖{a}A\setminus\{a\}. Estimate F(N)F(N). The lower bound F(N)≫N1/5F(N)\gg N^{1/5} is classical, from constructions of Erdős and Csaba, and every non-dividing set is non-averaging, which gave F(N)≤N1/4+o(1)F(N)\leq N^{1/4+o(1)}. The claimed new result is the matching upper bound F(N)≤N1/5+o(1)F(N)\leq N^{1/5+o(1)}, obtained by running the Pham-Zakharov density-increment argument one dimension lower through a projective normalization, hence F(N)=N1/5+o(1)F(N)=N^{1/5+o(1)}.
number theoryA0680632 sources
declared openerdos:176
Let N(k,ℓ)N(k, \ell) be the least NN such that every f:[N]→{−1,1}f : [N] \to \{-1, 1\} has a kk-term arithmetic progression PP with ∣∑n∈Pf(n)∣≥ℓ|\sum_{n \in P} f(n)| \ge \ell. In particular, is N(k,2)≤CkN(k, 2) \le C^k?
additive combinatoricsarithmetic progressionsdiscrepancypossible3 sources
declared openerdos:336
If h(r)h(r) is the maximal finite exact order attainable by an additive basis of order at most rr, what is lim⁡r→∞h(r)/r2\lim_{r \to \infty} h(r)/r^2? The candidate proof identifies the sharp limit 1/31/3.
number theoryadditive basispossible2 sources
declared openerdos:451
Let nkn_k be the least integer greater than 2k2k for which ∏i=1k(nk−i)\prod_{i=1}^k (n_k - i) has no prime factor in (k,2k)(k, 2k). How rapidly must nkn_k grow?
number theoryA3866203 sources
declared openerdos:514
For a transcendental entire function, how fast can ∣f(z)∣|f(z)| be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus M(r,f)M(r, f)?
analysisN/A4 sources
declared openerdos:550
Let m1≤⋯≤mkm_1\leq\cdots\leq m_k and nn be sufficiently large. If TT is a tree on nn vertices and GG is the complete multipartite graph with vertex class sizes m1,…,mkm_1,\ldots,m_k, prove that R(T,G)≤(χ(G)−1)(R(T,Km1,m2)−1)+m1R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.
graph theoryramsey theoryN/A4 sources
declared openerdos:584
Must every graph with nn vertices and δn2\delta n^2 edges contain large subgraphs in which every two edges lie on specified short cycles? A dense high-girth construction refutes the statement when δ\delta may shrink with nn.
graph theorycyclesN/A2 sources
declared openerdos:654
If nn planar points have no four concyclic, must some point determine (1−o(1))n(1 - o(1))n distinct distances? Failing that, can one always force more than (1/3+c)n(1/3 + c)n?
geometrydistancespossible3 sources
declared openerdos:662
Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and strict-shell readings.
geometrydistancesN/A3 sources
declared openerdos:670
Erdős asked whether every nn-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least (1+o(1))n2(1+o(1))n^2. Disproved: an explicit high-dimensional construction beats the conjectured constant.
geometrydistancesN/A2 sources
declared open · $250erdos:671
For triangular arrays of nodes ain∈[−1,1]a_i^n\in[-1,1] let Lnf\mathcal{L}^nf be the Lagrange interpolation polynomials of a continuous ff, with fundamental polynomials pinp_i^n. Is there a choice of nodes such that for every continuous ff there is some xx where lim sup⁡n∑i∣pin(x)∣=∞\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty and yet Lnf(x)→f(x)\mathcal{L}^nf(x) \to f(x)? Is there a choice with lim sup⁡n∑i∣pin(x)∣=∞\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty for every xx, yet for every continuous ff some xx has Lnf(x)→f(x)\mathcal{L}^nf(x)\to f(x)? Both questions are claimed resolved in the affirmative.
analysisN/A3 sources
declared openerdos:684
For the least kk at which the small-prime part of (nk)\binom{n}{k} exceeds n2n^2, how large can f(n)f(n) be?
number theoryprimesbinomial coefficientsA392019 · possible3 sources
declared openerdos:709
How long must an interval be to contain distinct representatives xix_i, with ai∣xia_i \mid x_i, for every nn-element set of moduli A={a1,…,an}A = \{a_1, \dots, a_n\}?
number theorypossible2 sources
declared openerdos:731
Let A(n)A(n) be the least positive integer not dividing (2nn)\binom{2n}{n}. Erdos asked for the behaviour of A(n)A(n) for reasonable nn. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals against the scale FX=2(log⁡2)1/4L1/4exp⁡(log⁡2)LF_X = \sqrt{2}(\log 2)^{1/4} L^{1/4} \exp\sqrt{(\log 2)L} with L=log⁡(2X)L = \log(2X).
number theorybinomial coefficientsA0061974 sources
declared openerdos:768
Let A(x)A(x) count n≤xn \le x such that every prime p∣np \mid n has a divisor d>1d > 1 of nn with d≡1(modp)d \equiv 1 \pmod p. Erdos asked whether A(x)/x=exp⁡(−(c+o(1))log⁡xlog⁡log⁡x)A(x)/x = \exp(-(c+o(1))\sqrt{\log x}\log\log x). It does, with c=1/(2log⁡2)c = 1/(2\sqrt{\log 2}).
number theoryA001034 · A3522873 sources
declared openerdos:856
Let k≥3k\geq 3 and fk(N)f_k(N) be the maximum of ∑n∈A1n\sum_{n\in A}\frac{1}{n} over all A⊆{1,…,N}A\subseteq\{1,\ldots,N\} containing no kk subsets with the same pairwise least common multiple. Estimate fk(N)f_k(N). The claimed answer: fk(N)=(log⁡N)γk+o(1)f_k(N)=(\log N)^{\gamma_k+o(1)}, where γk\gamma_k is a weighted generalization of the Tang-Zhang sunflower capacity.
number theorypossible3 sources
declared openerdos:866
Estimate the least excess gk(N)g_k(N) forcing kk integers whose pairwise sums all lie in a dense subset of {1,…,2N}\{1, \dots, 2N\}; in particular, determine the positive variant h4(n)h_4(n).
number theoryadditive combinatoricspossible5 sources
declared openerdos:870
Let k≥3k\geq 3 and AA be an additive basis of order kk. Does there exist a constant c=c(k)>0c=c(k)>0 such that if r(n)≥clog⁡nr(n)\geq c\log n for all large nn (where r(n)r(n) counts representations of nn as a sum of at most kk elements of AA) then AA must contain a minimal basis of order kk? The claimed answer is no, for every k≥3k\geq 3.
number theoryadditive basisN/A3 sources
declared openerdos:953
What is the largest possible measure of a subset of a radius-RR disk in R2\mathbb{R}^2 containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives M(R)≪R1/2M(R) \ll R^{1/2}; with Sárközy's lower construction, M(R)=R1/2+o(1)M(R) = R^{1/2 + o(1)}.
geometrydistancesN/A3 sources
declared openerdos:1032
Do arbitrarily large 4-chromatic edge-critical graphs exist with minimum degree bounded below by a positive constant times the number of vertices?
graph theorychromatic numberpossible3 sources
declared openerdos:1039
For f(z)=∏i=1n(z−zi)f(z) = \prod_{i=1}^n (z - z_i) with all ∣zi∣≤1|z_i| \le 1, let ρ(f)\rho(f) be the radius of the largest disc contained in {z:∣f(z)∣<1}\{z : |f(z)| < 1\}. Is ρ(f)≫1/n\rho(f) \gg 1/n? The worst case is now known to be Θ(1/n)\Theta(1/n), with the explicit bound ρ(f)≥(log⁡2)/n\rho(f) \ge (\log 2)/n.
analysisN/A3 sources
declared openerdos:1040
For a closed infinite set F⊆CF \subseteq \mathbb{C}, let μ(F)\mu(F) be the infimum of ∣{z:∣f(z)∣<1}∣|\{z : |f(z)| < 1\}| over monic polynomials with zeros in FF. Is μ(F)\mu(F) determined only by the transfinite diameter of FF?
analysisN/A3 sources
declared openerdos:1151
Let Lnf\mathcal{L}^nf be the Lagrange interpolation polynomials of a continuous ff on the Chebyshev nodes. Prove that, for any closed A⊆[−1,1]A\subseteq [-1,1], there exists a continuous function ff such that AA is the set of limit points of Lnf(x)\mathcal{L}^nf(x).
analysispolynomialsN/A3 sources
declared openerdos:1177
For a finite forbidden triple system GG, what exact uncountable chromatic cardinalities occur among GG-free triple systems, and how do those spectra interact? The revised manuscript answers the three exact-cardinal questions and claims a complete spectrum dichotomy.
set theorychromatic numberhypergraphsN/A3 sources
declared openerdos:1186
What is the minimum asymptotic density δk\delta_k of monochromatic kk-term arithmetic progressions in every two-colouring of {1,…,n}\{1, \dots, n\}? The exact certificate gives δ3=117/2192\delta_3 = 117/2192, matching the known 548-bead colouring.
additive combinatoricsarithmetic progressionspossible2 sources
declared openerdos:1189
For irreducible covering sets of size kk, determine their count, the possible largest modulus, the maximal reciprocal sum, and whether divisor-set examples occur infinitely often.
number theorycovering systemspossible2 sources

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