9 papers
Three-color van der Waerden numbers grow super-exponentially
Jacob Fox, Zach Hunter
For sufficiently large, we show that there is a three-coloring of the first positive integers without any monochromatic -term arithmetic progressions. T…
A note on the Alon-Saks-Seymour problem
Jacob Fox
Let be the maximum possible chromatic number of a graph whose edge set can be partitioned into at most complete bipartite graphs. Alon, Saks, and Seymour conjectured tha…
Finding blowups one vertex at a time
Jacob Fox, Yuval Wigderson, Yunkun Zhou
An influential theorem of Nikiforov states that if an -vertex graph contains at least copies of some fixed -vertex graph , then contains an -blowup of or…
Separators for intersection graphs of spheres
Jacob Fox, Jonathan Tidor
We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension . One of our results is that if an intersection graph of spheres i…
Coloring small locally sparse degenerate graphs and related problems
Domagoj Bradač, Jacob Fox, Raphael Steiner +2
The classic upper bound on the chromatic number of -degenerate graphs is , shown to be tight by complete graphs. A natural question is whether this bound remains tight if o…
Color-avoiding directed paths in tournaments
Jacob Fox, Benny Sudakov, Yuval Wigderson
We study the following Ramsey-theoretic question: given a -coloring of the edges of a tournament, how long of a directed path can we guarantee whose edges avoid one of the color…
On off-diagonal hypergraph Ramsey numbers
David Conlon, Jacob Fox, Benjamin Gunby +4
A fundamental problem in Ramsey theory is to determine the growth rate in terms of of the Ramsey number of a fixed -uniform hypergraph versus the compl…
Ramsey and Turán numbers of sparse hypergraphs
Jacob Fox, Maya Sankar, Michael Simkin +2
Degeneracy plays an important role in understanding Turán- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to captu…
The growth rate of multicolor Ramsey numbers of -graphs
Domagoj Bradač, Jacob Fox, Benny Sudakov
The -color Ramsey number of a -uniform hypergraph denoted , is the minimum integer such that any coloring of the edges of the complete -uniform hypergraph…