@article{19433, abstract = {An ordered r-matching is an r-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of r-dimensional orders. The theory of ordered 2-matchings is well developed and has connections and applications to extremal and enumerative combinatorics, probability and geometry. On the other hand, in the case r≥3 much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruciński made some first steps towards a general theory of ordered r-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain.}, author = {Anastos, Michael and Jin, Zhihan and Kwan, Matthew Alan and Sudakov, Benny}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Extremal, enumerative and probabilistic results on ordered hypergraph matchings}}, doi = {10.1017/fms.2024.144}, volume = {13}, year = {2025}, } @article{22158, abstract = {The triangle removal states that if G contains edge-disjoint triangles, then G contains triangles. Unfortunately, there are no sensible bounds on the order of growth of , and at any rate, it is known that is not polynomial in . Csaba recently obtained an asymmetric variant of the triangle removal, stating that if G contains edge-disjoint triangles, then G contains copies of . To this end, he devised a new variant of Szemerédi’s regularity lemma. We obtain the following results: • We first give a regularity-free proof of Csaba’s theorem, which improves the number of copies of to the optimal number . • We say that H is -abundant if every graph containing edge-disjoint triangles has copies of H. It is easy to see that a -abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are -abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative. Our proofs use a mix of combinatorial, number-theoretic, probabilistic and Ramsey-type arguments.}, author = {Gishboliner, Lior and Shapira, Asaf and Wigderson, Yuval}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{An efficient asymmetric removal lemma and its limitations}}, doi = {10.1017/fms.2024.68}, volume = {13}, year = {2025}, } @article{18107, abstract = {We consider a dilute fully spin-polarized Fermi gas at positive temperature in dimensions d∈{1,2,3} . We show that the pressure of the interacting gas is bounded from below by that of the free gas plus, to leading order, an explicit term of order adρ2+2/d, where a is the p-wave scattering length of the repulsive interaction and ρ is the particle density. The results are valid for a wide range of repulsive interactions, including that of a hard core, and uniform in temperatures at most of the order of the Fermi temperature. A central ingredient in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237–260).}, author = {Lauritsen, Asbjørn Bækgaard and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Pressure of a dilute spin-polarized Fermi gas: Lower bound}}, doi = {10.1017/fms.2024.56}, volume = {12}, year = {2024}, } @article{12148, abstract = {We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, keywords = {Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Theoretical Computer Science, Analysis}, publisher = {Cambridge University Press}, title = {{Rank-uniform local law for Wigner matrices}}, doi = {10.1017/fms.2022.86}, volume = {10}, year = {2022}, } @article{19491, abstract = {Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let D denote the set of positive squarefree integers having no prime factors congruent to 3 modulo 4 . Stevenhagen [19] conjectured that the density of d in D such that the negative Pell equation x2−dy2=−1 is solvable with x,y∈Z is 58.1% , to the nearest tenth of a percent. By studying the distribution of the 8 -rank of narrow class groups Cl+(d) of Q(√d) , we prove that the infimum of this density is at least 53.8% .}, author = {Chan, Yik Tung and Koymans, Peter and Milovic, Djordjo and Pagano, Carlo}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{The 8-rank of the narrow class group and the negative Pell equation}}, doi = {10.1017/fms.2022.40}, volume = {10}, year = {2022}, } @article{6355, abstract = {We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.}, author = {Akopyan, Arseniy and Avvakumov, Sergey}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Any cyclic quadrilateral can be inscribed in any closed convex smooth curve}}, doi = {10.1017/fms.2018.7}, volume = {6}, year = {2018}, }