8 papers
A note on arithmetic progressions with restricted differences
David Conlon, Jacob Fox, Huy Tuan Pham
In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricte…
On the clique number of random Cayley graphs and related topics
David Conlon, Jacob Fox, Huy Tuan Pham +1
We prove that a random Cayley graph on a group of order has clique number with high probability. This bound is best possible up to the constant factor f…
The largest subgraph without a forbidden induced subgraph
Jacob Fox, Rajko Nenadov, Huy Tuan Pham
We initiate the systematic study of the following Turán-type question. Suppose is a graph with vertices such that the edge density between any pair of subsets of vertices o…
Big line or big convex polygon
David Conlon, Jacob Fox, Xiaoyu He +3
Let be the minimum such that every -element point set in the plane contains either collinear members or points in convex position. We prove that th…
Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures
Matija Bucić, Jacob Fox, Huy Tuan Pham
It is well-known that polynomial versions of theorems of Rödl and Nikiforov, as conjectured by Fox and Sudakov and Nguyen, Scott and Seymour imply the classical Erdős-Hajnal conjec…
A question of Erdős and Graham on Egyptian fractions
David Conlon, Jacob Fox, Xiaoyu He +4
Answering a question of Erdős and Graham, we show that for each fixed positive rational number the number of ways to write as a sum of reciprocals of distinct positive inte…
A multipartite analogue of Dilworth's Theorem
Jacob Fox, Huy Tuan Pham
We prove that every partially ordered set on elements contains subsets such that either each of these subsets has size and, for every…
Homogeneous structures in subset sums and non-averaging sets
David Conlon, Jacob Fox, Huy Tuan Pham
We show that for every positive integer there are positive constants and such that if is a subset of of size at least , then, for so…