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1. COVERING INTEGERS BY $x^2 + dy^2$

   https://doi.org/10.1017/S1474748024000513  

   Green, Ben; Soundararajan, Kannan (May 2025, Journal of the Institute of Mathematics of Jussieu)

   Abstract What proportion of integers$$n \leq N$$may be expressed as$$x^2 + dy^2$$for some$$d \leq \Delta $$, with$$x,y$$integers? Writing$$\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$$for some$$\alpha \in (-\infty , \infty )$$, we show that the answer is$$\Phi (\alpha ) + o(1)$$, where$$\Phi $$is the Gaussian distribution function$$\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$$. A consequence of this is a phase transition: Almost none of the integers$$n \leq N$$can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 - \varepsilon }$$, but almost all of them can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$$. 

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2. Transversal C_k -factors in subgraphs of the balanced blow-up of C_k

   https://doi.org/10.1017/S0963548322000086  

   Ergemlidze, Beka; Molla, Theodore (November 2022, Combinatorics, Probability and Computing)

   Abstract For a subgraph$$G$$of the blow-up of a graph$$F$$, we let$$\delta ^*(G)$$be the smallest minimum degree over all of the bipartite subgraphs of$$G$$induced by pairs of parts that correspond to edges of$$F$$. Johansson proved that if$$G$$is a spanning subgraph of the blow-up of$$C_3$$with parts of size$$n$$and$$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$$, then$$G$$contains$$n$$vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$$G$$is a spanning subgraph of the blow-up of$$C_k$$with parts of size$$n$$and$$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$$, then$$G$$contains$$n$$vertex disjoint copies of$$C_k$$such that each$$C_k$$intersects each of the$$k$$parts exactly once. A similar conjecture was also made by Fischer and the case$$k=3$$was proved for large$$n$$by Magyar and Martin. In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of$$G$$to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically. 

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3. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

   Jiang, Tao; Longbrake, Sean; Ma, Jie (February 2023, Combinatorics, Probability and Computing)

   Abstract Given a family$$\mathcal{F}$$of bipartite graphs, theZarankiewicz number$$z(m,n,\mathcal{F})$$is the maximum number of edges in an$$m$$by$$n$$bipartite graph$$G$$that does not contain any member of$$\mathcal{F}$$as a subgraph (such$$G$$is called$$\mathcal{F}$$-free). For$$1\leq \beta \lt \alpha \lt 2$$, a family$$\mathcal{F}$$of bipartite graphs is$$(\alpha,\beta )$$-smoothif for some$$\rho \gt 0$$and every$$m\leq n$$,$$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$$(\alpha,\beta )$$-smooth family$$\mathcal{F}$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$$is bipartite. In this paper, we strengthen their result by showing that for every real$$\delta \gt 0$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\delta n^{\alpha -1}$$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$$\mathcal{F}$$consisting of the single graph$$K_{s,t}$$when$$t\gg s$$. We also prove an analogous result for$$C_{2\ell }$$-free graphs for every$$\ell \geq 2$$, which complements a result of Keevash, Sudakov and Verstraëte. 

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4. Counting independent sets in structured graphs

   https://doi.org/10.1017/S0963548325000124  

   Bucić, Matija; Chudnovsky, Maria; Codsi, Julien (July 2025, Combinatorics, Probability and Computing)

   Abstract Counting independent sets in graphs and hypergraphs under a variety of restrictions is a classical question with a long history. It is the subject of the celebrated container method which found numerous spectacular applications over the years. We consider the question of how many independent sets we can have in a graph under structural restrictions. We show that any$$n$$-vertex graph with independence number$$\alpha$$without$$bK_a$$as an induced subgraph has at most$$n^{O(1)} \cdot \alpha ^{O(\alpha )}$$independent sets. This substantially improves the trivial upper bound of$$n^{\alpha },$$whenever$$\alpha \le n^{o(1)}$$and gives a characterisation of graphs forbidding which allows for such an improvement. It is also in general tight up to a constant in the exponent since there exist triangle-free graphs with$$\alpha ^{\Omega (\alpha )}$$independent sets. We also prove that if one in addition assumes the ground graph is chi-bounded one can improve the bound to$$n^{O(1)} \cdot 2^{O(\alpha )}$$which is tight up to a constant factor in the exponent. 

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5. Higher uniformity of arithmetic functions in short intervals I. All intervals

   https://doi.org/10.1017/fmp.2023.28  

   Matomäki, Kaisa; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni (January 2023, Forum of Mathematics, Pi)

   Abstract We study higher uniformity properties of the Möbius function$$\mu $$, the von Mangoldt function$$\Lambda $$, and the divisor functions$$d_k$$on short intervals$$(X,X+H]$$with$$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$$for a fixed constant$$0 \leq \theta < 1$$and any$$\varepsilon>0$$. More precisely, letting$$\Lambda ^\sharp $$and$$d_k^\sharp $$be suitable approximants of$$\Lambda $$and$$d_k$$and$$\mu ^\sharp = 0$$, we show for instance that, for any nilsequence$$F(g(n)\Gamma )$$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when$$\theta = 5/8$$and$$f \in \{\Lambda , \mu , d_k\}$$or$$\theta = 1/3$$and$$f = d_2$$. As a consequence, we show that the short interval Gowers norms$$\|f-f^\sharp \|_{U^s(X,X+H]}$$are also asymptotically small for any fixedsfor these choices of$$f,\theta $$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$$L^2$$. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$$II$$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$$I_2$$sums. 

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