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1. 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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2. Weighted estimates for one-sided martingale transforms

   https://doi.org/10.1090/proc/14665  

   Chen, Wei; Han, Rui; Lacey, Michael T. (October 2019, Proceedings of the American Mathematical Society)

   Let $$ Tf =\sum _{I} \varepsilon _I \langle f,h_{I^+}\rangle h_{I^-}$$. Here, $$ \lvert \varepsilon _I\rvert =1 $$, and $$ h_J$$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, $$\displaystyle \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} .$$ Above, we use the one-sided $$ A_2$$ characteristic for the weight $ w$. This is an instance of a one-sided $$ A_2$$ conjecture. Our proof of this fact is difficult, as the very quick known proofs of the $$ A_2$$ theorem do not seem to apply in the one-sided setting. 

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3. Removal lemmas and approximate homomorphisms

   https://doi.org/10.1017/S0963548321000572  

   Fox, Jacob; Zhao, Yufei (July 2022, Combinatorics, Probability and Computing)

   Abstract We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma , states that for each $$\epsilon>0$$ there exists M such that every triangle-free graph G has an $$\epsilon$$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $$\epsilon$$ -approximate homomorphism is a map $$V(G) \to V(F)$$ where all but at most $$\epsilon \left\lvert{V(G)}\right\rvert^2$$ edges of G are mapped to edges of F ). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $$\epsilon^{-1}$$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal. 

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4. Convergence of graph Laplacian with kNN self-tuned kernels

   https://doi.org/10.1093/imaiai/iaab019  

   Cheng, Xiuyuan; Wu, Hau-Tieng (September 2021, Information and Inference: A Journal of the IMA)

   Abstract Kernelized Gram matrix $$W$$ constructed from data points $$\{x_i\}_{i=1}^N$$ as $$W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma ^2} ) $$ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $$\sigma $$, and a common practice called self-tuned kernel adaptively sets a $$\sigma _i$$ at each point $$x_i$$ by the $$k$$-nearest neighbor (kNN) distance. When $$x_i$$s are sampled from a $$d$$-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $$L_N$$ to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels $$W^{(\alpha )}_{ij} = k_0( \frac{ \| x_i - x_j \|^2}{ \epsilon \hat{\rho }(x_i) \hat{\rho }(x_j)})/\hat{\rho }(x_i)^\alpha \hat{\rho }(x_j)^\alpha $$, where $$\hat{\rho }$$ is the estimated bandwidth function by kNN and the limiting operator is also parametrized by $$\alpha $$. When $$\alpha = 1$$, the limiting operator is the weighted manifold Laplacian $$\varDelta _p$$. Specifically, we prove the point-wise convergence of $$L_N f $$ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$ consistency for $$\hat{\rho }$$ which bounds the relative estimation error $$|\hat{\rho } - \bar{\rho }|/\bar{\rho }$$ uniformly with high probability, where $$\bar{\rho } = p^{-1/d}$$ and $$p$$ is the data density function. Our theoretical results reveal the advantage of the self-tuned kernel over the fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of $$d$$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data. 

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5. Balanced Allocation Through Random Walk

   Frieze, A.; Petti, S (December 2018, Information processing letters)

   We consider the allocation problem in which $$m \leq (1-\epsilon) dn $$ items are to be allocated to $$n$$ bins with capacity $$d$$. The items $$x_1,x_2,\ldots,x_m$$ arrive sequentially and when item $$x_i$$ arrives it is given two possible bin locations $$p_i=h_1(x_i),q_i=h_2(x_i)$$ via hash functions $$h_1,h_2$$. We consider a random walk procedure for inserting items and show that the expected time insertion time is constant provided $$\epsilon = \Omega\left(\sqrt{ \frac{ \log d}{d}} \right).$$ 

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