More Like this

1. On a problem of M. Talagrand

   https://doi.org/10.1002/rsa.21077  

   Frankston, Keith; Kahn, Jeff; Park, Jinyoung (December 2022, Random Structures & Algorithms)

   Abstract We address a special case of a conjecture of M. Talagrand relating two notions of “threshold” for an increasing family of subsets of a finite setV. The full conjecture implies equivalence of the “Fractional Expectation‐Threshold Conjecture,” due to Talagrand and recently proved by the authors and B. Narayanan, and the (stronger) “Expectation‐Threshold Conjecture” of the second author and G. Kalai. The conjecture under discussion here says there is a fixedLsuch that if, for a given , admits withand(a.k.a.  isweakly p‐small), then admits such a taking values in ( is‐small). Talagrand showed this when is supported on singletons and suggested, as a more challenging test case, proving it when is supported on pairs. The present work provides such a proof. 

   more » « less
2. Big polynomial rings and Stillman's Conjecture

   https://doi.org/10.1007/s00222-019-00889-y  

   Erman, Daniel; Sam, Steven V; Snowden, Andrew (May 2019, Inventiones mathematicae)

   The purpose of this paper is to prove that certain limits of polynomial rings are themselves polynomial rings, and show how this observation can be used to deduce some interesting results in commutative algebra. In particular, we give two new proofs of Stillman's conjecture. The first is similar to that of Ananyan-Hochster, though more streamlined; in particular, it establishes the existence of small subalgebras. The second proof is completely different, and relies on a recent noetherianity result of Draisma. 

   more » « less
3. Induced subgraph density. VI. Bounded VC-dimension

   https://doi.org/10.1016/j.aim.2025.110601  

   Nguyen, Tung; Scott, Alex; Seymour, Paul (December 2025, Advances in Mathematics)

   We confirm a conjecture of Fox, Pach, and Suk, that for every d>0, there exists c>0 such that every n-vertex graph of VC-dimension at most d has a clique or stable set of size at least nc . This implies that, in the language of model theory, every graph definable in NIP structures has a clique or anti-clique of polynomial size, settling a conjecture of Chernikov, Starchenko, and Thomas. Our result also implies that every two-colourable tournament satisfies the tournament version of the Erdős-Hajnal conjecture, which completes the verification of the conjecture for six-vertex tournaments. The result extends to uniform hypergraphs of bounded VC-dimension as well. The proof method uses the ultra-strong regularity lemma for graphs of bounded VC-dimension proved by Lovász and Szegedy and the method of iterative sparsification introduced by the authors in an earlier paper. 

   more » « less
4. Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes

   https://doi.org/10.1007/s00208-023-02597-8  

   Mehlhop, Nathan; Słomian, Wojciech (March 2024, Mathematische Annalen)

   Abstract We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subsets of primes. This is a contribution to the Rosenblatt–Wierdl conjecture (Lond Math Soc Lect Notes 205:3–151, 1995, Problem 4.12, p. 80) with averages taken over primes. These inequalities provide endpoints for ther-variational estimates obtained by Trojan (Math Ann 374:1597–1656, 2019). 

   more » « less
5. Bounds on the dimension of lineal extensions

   https://doi.org/10.4171/JFG/161  

   Bushling, Ryan_E G; Fiedler, Jacob B (March 2025, Journal of Fractal Geometry, Mathematics of Fractals and Related Topics)

   LetE \subseteq \mathbb{R}^{n}be a union of line segments andF \subseteq \mathbb{R}^{n}the set obtained fromEby extending each line segment inEto a full line. Keleti’sline segment extension conjectureposits that the Hausdorff dimension ofFshould equal that ofE. Working in\mathbb{R}^{2}, we use effective methods to prove a strong packing dimension variant of this conjecture. Furthermore, a key inequality in this proof readily entails the planar case of the generalized Kakeya conjecture for packing dimension. This is followed by several doubling estimates in higher dimensions and connections to related problems. 

   more » « less