Joshua Hinman proved Bárány’s conjecture.
One of my first posts on this blog was a 2008 post Five Open Problems Regarding Convex Polytopes, now 14 years later, I can tell you about the first problem on the list to get solved.
Imre Bárány posed in the late 1990s the following question:
For a 
Now, Joshua Hinman settled the problem! In his paper A Positive Answer to Bárány’s Question on Face Numbers of Polytopes he actually proved even stronger linear relations. The abstract of Joshua’s paper starts with the very true assertion: “Despite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood.” He moved on to describe his new inequalities:
![\frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} + {\lfloor \frac{d}{2} \rfloor \choose k}\biggr], \qquad \frac{f_k(P)}{f_{d-1}(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose d-k-1} + {\lfloor \frac{d}{2} \rfloor \choose d-k-1}\biggr].](https://s0.wp.com/latex.php?latex=%5Cfrac%7Bf_k%28P%29%7D%7Bf_0%28P%29%7D+%5Cgeq+%5Cfrac%7B1%7D%7B2%7D%5Cbiggl%5B%7B%5Clceil+%5Cfrac%7Bd%7D%7B2%7D+%5Crceil+%5Cchoose+k%7D+%2B+%7B%5Clfloor+%5Cfrac%7Bd%7D%7B2%7D+%5Crfloor+%5Cchoose+k%7D%5Cbiggr%5D%2C+%5Cqquad+%5Cfrac%7Bf_k%28P%29%7D%7Bf_%7Bd-1%7D%28P%29%7D+%5Cgeq+%5Cfrac%7B1%7D%7B2%7D%5Cbiggl%5B%7B%5Clceil+%5Cfrac%7Bd%7D%7B2%7D+%5Crceil+%5Cchoose+d-k-1%7D+%2B+%7B%5Clfloor+%5Cfrac%7Bd%7D%7B2%7D+%5Crfloor+%5Cchoose+d-k-1%7D%5Cbiggr%5D.&bg=ffffff&fg=333333&s=0&c=20201002)
Lei Xue proved Grünbaum’s conjecture
In her 2020 paper: A Proof of Grünbaum’s Lower Bound Conjecture for general polytopes, Lei Xue proved a lower bound conjecture of Grünbaum: In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s≤2d vertices has at least
k-faces. Lei Xue proved this conjecture and also characterized the cases in which equality holds.
Congratulations to Lei Xue and to Joshua Hinman.
Combinatorics and more 
Can it help shed some light on your 3^d conjecture?
I dont know.
Very nice. Are you going to blog about Park and Pham’s proof of the Kahn-Kalai conjecture?
Dear kodlu, I certainly should blog about the proof! (And I am also thinking about blogging about our old 2006 program toward a proof and some related open questions.)