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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. Zarankiewicz’s problem for semilinear hypergraphs

   https://doi.org/10.1017/fms.2021.52  

   Basit, Abdul; Chernikov, Artem; Starchenko, Sergei; Tao, Terence; Tran, Chieu-Minh (January 2021, Forum of Mathematics, Sigma)

   Abstract A bipartite graph $$H = \left (V_1, V_2; E \right )$$ with $$\lvert V_1\rvert + \lvert V_2\rvert = n$$ is semilinear if $$V_i \subseteq \mathbb {R}^{d_i}$$ for some $$d_i$$ and the edge relation E consists of the pairs of points $$(x_1, x_2) \in V_1 \times V_2$$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $$d_1 + d_2$$ variables for some s . We show that for a fixed k , the number of edges in a $$K_{k,k}$$ -free semilinear H is almost linear in n , namely $$\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ for any $$\varepsilon> 0$$ ; and more generally, $$\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$$ for a $$K_{k, \dotsc ,k}$$ -free semilinear r -partite r -uniform hypergraph. As an application, we obtain the following incidence bound: given $$n_1$$ points and $$n_2$$ open boxes with axis-parallel sides in $$\mathbb {R}^d$$ such that their incidence graph is $$K_{k,k}$$ -free, there can be at most $$O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o -minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner). 

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