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1. On Off-Diagonal Hypergraph Ramsey Numbers

   https://doi.org/10.1093/imrn/rnaf122  

   Conlon, David; Fox, Jacob; Gunby, Benjamin; He, Xiaoyu; Mubayi, Dhruv; Suk, Andrew; Verstraëte, Jacques (June 2025, International Mathematics Research Notices)

   Abstract A fundamental problem in Ramsey theory is to determine the growth rate in terms of $$n$$ of the Ramsey number $$r(H, K_{n}^{(3)})$$ of a fixed $$3$$-uniform hypergraph $$H$$ versus the complete $$3$$-uniform hypergraph with $$n$$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $$H$$, including links of odd cycles and tight cycles of length not divisible by three, $$r(H, K_{n}^{(3)}) \ge 2^{\Omega _{H}(n \log n)}$$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $$H$$ for which $$r(H, K_{n}^{(3)})$$ is superpolynomial in $$n$$. This provides the first example of a separation between $$r(H,K_{n}^{(3)})$$ and $$r(H,K_{n,n,n}^{(3)})$$, since the latter is known to be polynomial in $$n$$ when $$H$$ is linear. 

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2. On Explicit Constructions of Designs

   https://doi.org/10.37236/10513  

   Liu, Xizhi; Mubayi, Dhruv (January 2022, The Electronic Journal of Combinatorics)

   An $(n,r,s)$-system is an $$r$$-uniform hypergraph on $$n$$ vertices such that every pair of edges has an intersection of size less than $$s$$. Using probabilistic arguments, Rödl and Šiňajová showed that for all fixed integers $$r> s \ge 2$$, there exists an $(n,r,s)$-system with independence number $$O\left(n^{1-\delta+o(1)}\right)$$ for some optimal constant $$\delta >0$$ only related to $$r$$ and $$s$$. We show that for certain pairs $(r,s)$ with $$s\le r/2$$ there exists an explicit construction of an $(n,r,s)$-system with independence number $$O\left(n^{1-\epsilon}\right)$$, where $$\epsilon > 0$$ is an absolute constant only related to $$r$$ and $$s$$. Previously this was known only for $s>r/2$ by results of Chattopadhyay and Goodman. 

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3. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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4. More on zeros and approximation of the Ising partition function

   https://doi.org/10.1017/fms.2021.40  

   Barvinok, Alexander; Barvinok, Nicholas (January 2021, Forum of Mathematics, Sigma)

   Abstract We consider the problem of computing the partition function $$\sum _x e^{f(x)}$$ , where $$f: \{-1, 1\}^n \longrightarrow {\mathbb R}$$ is a quadratic or cubic polynomial on the Boolean cube $$\{-1, 1\}^n$$ . In the case of a quadratic polynomial f , we show that the partition function can be approximated within relative error $$0 < \epsilon < 1$$ in quasi-polynomial $$n^{O(\ln n - \ln \epsilon )}$$ time if the Lipschitz constant of the non-linear part of f with respect to the $$\ell ^1$$ metric on the Boolean cube does not exceed $$1-\delta $$ , for any $$\delta>0$$ , fixed in advance. For a cubic polynomial f , we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that $$\sum _x e^{\tilde {f}(x)} \ne 0$$ for complex-valued polynomials $$\tilde {f}$$ in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee–Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices. 

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5. THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

   https://doi.org/10.1017/S1474748019000318  

   Corvaja, Pietro; Ghioca, Dragos; Scanlon, Thomas; Zannier, Umberto (March 2021, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract Let $$K$$ be an algebraically closed field of prime characteristic $$p$$ , let $$X$$ be a semiabelian variety defined over a finite subfield of $$K$$ , let $$\unicode[STIX]{x1D6F7}:X\longrightarrow X$$ be a regular self-map defined over $$K$$ , let $$V\subset X$$ be a subvariety defined over $$K$$ , and let $$\unicode[STIX]{x1D6FC}\in X(K)$$ . The dynamical Mordell–Lang conjecture in characteristic $$p$$ predicts that the set $$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$$ is a union of finitely many arithmetic progressions, along with finitely many $$p$$ -sets, which are sets of the form $$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$$ for some $$m\in \mathbb{N}$$ , some rational numbers $$c_{i}$$ and some non-negative integers $$k_{i}$$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $$X$$ is an algebraic torus, we can prove the conjecture in two cases: either when $$\dim (V)\leqslant 2$$ , or when no iterate of $$\unicode[STIX]{x1D6F7}$$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $$X$$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction. 

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