13 papers
A new lower bound for two-color van der Waerden numbers
Marcelo Campos, Jacob Fox, Carl Schildkraut
The van der Waerden number is the smallest positive integer such that every two-coloring of contains a monochromatic -term arithmetic progression.…
On nearly consecutive sequences without long arithmetic progressions
Jacob Fox, Carl Schildkraut
A sequence of integers is nearly consecutive if for . We prove that there are nearly consecutive sequences of length $…
More sum-product type counterexamples: products with shifts and
Oliver Roche-Newton, Carl Schildkraut, Audie Warren
Adapting the construction disproving the sum-product conjecture over present in Bloom, Sawin, Schildkraut and Zhelezov, we show the existence of a constant and ar…
Dirac subgraphs of powers of cycles are Hamiltonian
Richard Lang, Alp Müyesser, Mathias Schacht +1
We show that, for every and all sufficiently large , any spanning subgraph of the th power of a cycle with minimum degree at least contains…
The sum-product conjecture is false for real numbers
Thomas F Bloom, Will Sawin, Carl Schildkraut +1
We disprove the sum-product conjecture for real numbers by constructing arbitrarily large (whose elements are algebraic integers in a number field of degree $…
Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the ErdÅs Problems
Tony Feng, Trieu Trinh, Garrett Bingham +21
We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's ErdÅs Problems database. We emp…