Related papers: Hjorth's reflection argument

We show that there are many (compact) convex semi-algebraic sets in euclidean space that do not have a semidefinite representation. This gives a negative answer to a question by Nemirovski, resp. it shows that the Helton-Nie conjecture is…

Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. P\'alfy asserts that the complement…

We prove that the set of exceptional $\lambda\in (1/2,1)$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the…

Erd\"os conjectured the existence of an infinite Sidon sequence of positive integers which is also an asymptotic basis of order 3. We make progress towards this conjecture in several directions. First we prove the conjecture for all cyclic…

In this paper we show that for every $2\leq n\in \mathbb{N}$, the statement "there is an $n$-entangled set, but there are no $n+1$-entangled sets" is consistent. We also prove some theorems which improve our understanding of entangled sets…

The core of a finite-dimensional modular representation $M$ of a finite group $G$ is its largest non-projective summand. We prove that the dimensions of the cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic,…

Combining stationary reflection (a compactness property) with the failure of SCH (an instance of non-compactness) has been a long-standing theme. We obtain this at $\aleph_{\omega_1}$, answering a question of Ben-Neria, Hayut, and Unger: We…

We provide a cyclic permutation analogue of the Erd\H os-Szekeres theorem. In particular, we show that every cyclic permutation of length $(k-1)(\ell-1)+2$ has either an increasing cyclic sub-permutation of length $k+1$ or a decreasing…

The Hofstadter $H$ sequence is defined by $H(1) = 1$ and $H(n) = n-H(H(H(n-1)))$ for $n > 1$. If $\alpha$ is the real root of $x^3+x=1$ we show that the numbers $\alpha H(n) \mod 1$ are not uniformly distributed on $[0,1]$, but converge to…

It is consistent that for every n >= 2, every stationary subset of omega_n consisting of ordinals of cofinality omega_k where k = 0 or k <= n-3 reflects fully in the set of ordinals of cofinality omega_{n-1}. We also show that this result…

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic…

We settle 22 conjectures of Cohen about cyclic numbers (positive integers $n$ with $\gcd(n,\varphi(n))=1$), proving 16 and disproving 6, and we completely resolve a related OEIS problem about sequences whose running averages are Fibonacci…

The Coleman-Oort conjecture says that for large $g$ there are no positive-dimensional Shimura subvarieties of $\mathsf{A}_g$ generically contained in the Jacobian locus. Counterexamples are known for $g\leq 7$. They can all be constructed…

We show that sequences of positive integers whose ratios $a_n^2/a_{n+1}$ lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only…

An analysis is given of $*$-representations of rank 2 single vertex graphs. We develop dilation theory for the non-selfadjoint algebras $\A_\theta$ and $\A_u$ which are associated with the commutation relation permutation $\theta$ of a 2…

We investigate the set of Pi-1-2 sentences which are Pi-1-1 conservative over the theories of reverse mathematics RCA0+ISigma_n and ACA0. We exhibit new elements of these sets and conclude that the sets are Pi_2 complete. Along the way, we…

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite…

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge n/k$…

We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending ${\sf RCA}_0$ and axiomatizable by a…

We prove that a strong version of Chang's Conjecture, equivalent to the Weak Reflection Principle at $\omega_2$, together with $2^\omega=\omega_2$, imply there are no $\omega_2$-Aronszajn trees.