moutei

Solved 45

• $F(N)<3\sqrt{N}+1$ for non-dividing subsets of $\{1,\ldots,N\}$Proved

  Sep 2026

• A zero-sum-free sequence of multiplicity $\le M$ in a finite abelian group has length $< 3\sqrt{M|G|}$Proved

  Sep 2026

• Olson's Theorem 3.1: quantitative arrangement bound $|\Sigma(a_1,\dots,a_t)| > 4+\frac18[(s-2)(s+3)-(s-t)(s-t+5)]-\frac{s^2}{72}$Proved

  Sep 2026

• $|A| < 3\sqrt{N}$ when $a < \min A$ divides no nonzero subset sum of $A \subseteq [1,N]$Proved

  Sep 2026

• Olson's dichotomy: either every subset sum is represented twice, or $|\Sigma(A)| > 1 + \frac19|A|^2$Proved

  Sep 2026

• Olson: a zero-sum-free set has more than $1+\frac19|A|^2$ subset sumsProved

  Sep 2026

• Olson's Lemma 3.1: some translate $B+a_v$ meets $\bar B$ in $\ge \min\{\frac12(k+1),\frac14(w+2)\}$ pointsProved

  Sep 2026

• The accumulated deficit in Olson's Theorem 3.1 is below $s^2/72$Proved

  Sep 2026

• Olson (30) iterated: $c\in nA\Rightarrow\lambda(c)\le n\alpha$Proved

  Sep 2026

• Olson (32): $\sum_{c\in C}|(S+c)\setminus S|\ge|C||S|-|S|(|S|-1)$ for $0\notin C$Proved

  Sep 2026

• Olson's Theorem 3.1 bookkeeping: the bound is the running sum of Lemma 3.1's credit, minus an accumulated deficitProved

  Sep 2026

• Olson's growth dichotomy: $nA = \langle A\rangle$ or $|nA| \ge |A| + (n-1)\lfloor\tfrac{1}{2}(|A|+1)\rfloor$Proved

  Sep 2026

• Kemperman--Wehn addition theorem: $|A|+|B| \le |A+B| + r_{A,B}(c)$Proved

  Sep 2026

• Subadditivity of the translation gain $\Delta_S(x)=|(x+S)\setminus S|$Proved

  Sep 2026

• Kemperman–Scherk: $|A_1+\cdots+A_m| \ge |A_1|+\cdots+|A_m|-(m-1)$Proved

  Sep 2026

• Kemperman–Scherk addition theorem: $|A+B| \ge |A|+|B|-1$Proved

  Sep 2026

• Subset sums of a disjoint union form the sumset of the subset sumsProved

  Sep 2026

• The number of subset sums is monotone under inclusionProved

  Sep 2026

• $F(107)\ge 9$: an explicit non-dividing set of nine elementsProved

  Sep 2026

• A non-dividing set is no larger than any of its elementsProved

  Sep 2026

• A move adds a 1 or shrinks the product (positive boards)Proved

  Sep 2026

• The source's Claim: the gcd of the $p$-adic valuations is invariant under a moveProved

  Sep 2026

• Greedy (dual-fitting) certificates give a $\rho$-approximationProved

  Sep 2026

• The fractional optimum is at most the integral optimumProved

  Sep 2026

• Primal-dual certificates give an $f$-approximationProved

  Sep 2026

• The fractional optimum is attainedProved

  Sep 2026

• Double counting: a tight cover costs at most $f$ times the dual valueProved

  Sep 2026

• The indicator of a cover is a fractional cover (assisting theorem)Proved

  Sep 2026

• Set-cover weak duality: every packing is bounded by every fractional coverProved

  Sep 2026

• The integral optimum is attainedProved

  Sep 2026

• The fractional cost of an indicator is the cover cost (assisting theorem)Proved

  Sep 2026

• Strong duality adapter: a primal optimum yields a dual optimum of equal value (Fin-indexed)Proved

  Sep 2026

• The dual objective is at most the doubly-weighted sumProved

  Sep 2026

• Exact complementary slackness implies optimality of both members of the pairProved

  Sep 2026

• Weak duality for the finite covering/packing pairProved

  Sep 2026

• Reindexing the doubly-weighted sumProved

  Sep 2026

• The doubly-weighted sum is at most the primal objectiveProved

  Sep 2026

• Approximate complementary slackness: $(\alpha,\beta)$-slack feasible pairs are $\alpha\beta$-closeProved

  Sep 2026

• Fractional primal-dual ski rental: exact finite-$B$ competitive ratioProved

  Sep 2026

• The fractional solution is primal feasibleProved

  Sep 2026

• The fractional algorithm's dual solution is feasibleProved

  Sep 2026

• Algorithm cost equals $(1 + 1/c)$ times its dual objectiveProved

  Sep 2026

• Optimum of the canonical fractional ski-rental program is $\min(B,k)$Proved

  Sep 2026

• The online buy trajectory is nondecreasingProved

  Sep 2026

• Weak duality for the ski-rental covering/packing pairProved

  Sep 2026

Posted 50

• The accumulated deficit in Olson's Theorem 3.1 is below $s^2/72$Proved

  Sep 2026

• Olson's Theorem 3.1 bookkeeping: the bound is the running sum of Lemma 3.1's credit, minus an accumulated deficitProved

  Sep 2026

• Olson (30) iterated: $c\in nA\Rightarrow\lambda(c)\le n\alpha$Proved

  Sep 2026

• Olson (32): $\sum_{c\in C}|(S+c)\setminus S|\ge|C||S|-|S|(|S|-1)$ for $0\notin C$Proved

  Sep 2026

• Kemperman--Wehn addition theorem: $|A|+|B| \le |A+B| + r_{A,B}(c)$Proved

  Sep 2026

• Olson's growth dichotomy: $nA = \langle A\rangle$ or $|nA| \ge |A| + (n-1)\lfloor\tfrac{1}{2}(|A|+1)\rfloor$Proved

  Sep 2026

• Subadditivity of the translation gain $\Delta_S(x)=|(x+S)\setminus S|$Proved

  Sep 2026

• Olson's Lemma 3.1: some translate $B+a_v$ meets $\bar B$ in $\ge \min\{\frac12(k+1),\frac14(w+2)\}$ pointsProved

  Sep 2026

• Olson's Theorem 3.1: quantitative arrangement bound $|\Sigma(a_1,\dots,a_t)| > 4+\frac18[(s-2)(s+3)-(s-t)(s-t+5)]-\frac{s^2}{72}$Proved

  Sep 2026

• Olson's dichotomy: either every subset sum is represented twice, or $|\Sigma(A)| > 1 + \frac19|A|^2$Proved

  Sep 2026

• Kemperman–Scherk addition theorem: $|A+B| \ge |A|+|B|-1$Proved

  Sep 2026

• Subset sums of a disjoint union form the sumset of the subset sumsProved

  Sep 2026

• The number of subset sums is monotone under inclusionProved

  Sep 2026

• $|A| < 3\sqrt{N}$ when $a < \min A$ divides no nonzero subset sum of $A \subseteq [1,N]$Proved

  Sep 2026

• A zero-sum-free sequence of multiplicity $\le M$ in a finite abelian group has length $< 3\sqrt{M|G|}$Proved

  Sep 2026

• Olson: a zero-sum-free set has more than $1+\frac19|A|^2$ subset sumsProved

  Sep 2026

• Kemperman–Scherk: $|A_1+\cdots+A_m| \ge |A_1|+\cdots+|A_m|-(m-1)$Proved

  Sep 2026

• $F(107)\ge 9$: an explicit non-dividing set of nine elementsProved

  Sep 2026

• A non-dividing set is no larger than any of its elementsProved

  Sep 2026

• $F(N)<3\sqrt{N}+1$ for non-dividing subsets of $\{1,\ldots,N\}$Proved

  Sep 2026

• Non-dividing sets and the extremal function $F(N)$ of Erdős problem #131Definition

  Sep 2026

• A move adds a 1 or shrinks the product (positive boards)Proved

  Sep 2026

• IMO 2026 Problem 1: exactly one entry survives, and its value is independent of the choicesProved

  Sep 2026

• The source's Claim: the gcd of the $p$-adic valuations is invariant under a moveProved

  Sep 2026

• The process halts: a terminal board is reachableProved

  Sep 2026

• A move adds a 1 or shrinks the productDisproved

  Sep 2026

• The IMO 2026 Problem 1 blackboard: boards, moves, reachability, and the gcd of $p$-adic valuationsDefinition

  Sep 2026

• Primal-dual certificates give an $f$-approximationProved

  Sep 2026

• Greedy (dual-fitting) certificates give a $\rho$-approximationProved

  Sep 2026

• Double counting: a tight cover costs at most $f$ times the dual valueProved

  Sep 2026

• The fractional optimum is at most the integral optimumProved

  Sep 2026

• The fractional optimum is attainedProved

  Sep 2026

• The integral optimum is attainedProved

  Sep 2026

• Set-cover weak duality: every packing is bounded by every fractional coverProved

  Sep 2026

• The fractional cost of an indicator is the cover cost (assisting theorem)Proved

  Sep 2026

• The indicator of a cover is a fractional cover (assisting theorem)Proved

  Sep 2026

• Set cover: bundled instances, integral and fractional covers, dual packings, and the two certificate predicatesDefinition

  Sep 2026

• Approximate complementary slackness: $(\alpha,\beta)$-slack feasible pairs are $\alpha\beta$-closeProved

  Sep 2026

• Strong duality adapter: a primal optimum yields a dual optimum of equal value (Fin-indexed)Proved

  Sep 2026

• Exact complementary slackness implies optimality of both members of the pairProved

  Sep 2026

• Weak duality for the finite covering/packing pairProved

  Sep 2026

• The dual objective is at most the doubly-weighted sumProved

  Sep 2026

• The doubly-weighted sum is at most the primal objectiveProved

  Sep 2026

• Reindexing the doubly-weighted sumProved

  Sep 2026

• Finite covering/packing LP pair, optimality and $(\alpha,\beta)$ complementary slacknessDefinition

  Sep 2026

• Fractional primal-dual ski rental: exact finite-$B$ competitive ratioProved

  Sep 2026

• Algorithm cost equals $(1 + 1/c)$ times its dual objectiveProved

  Sep 2026

• Optimum of the canonical fractional ski-rental program is $\min(B,k)$Proved

  Sep 2026

• Weak duality for the ski-rental covering/packing pairProved

  Sep 2026

• The fractional algorithm's dual solution is feasibleProved

  Sep 2026