Sarosh Adenwalla

Boolean combinations allow combining given combinatorial objects to obtain new, potentially more complicated, objects. In this paper, we initiate a systematic study of this idea applied to graphs. In order to understand expressive power and limitations of boolean combinations in this context, we investigate how they affect different combinatorial and structural properties of graphs, in particular χ-boundedness, as well as characterize the structure of boolean combinations of graphs from various classes..

Erdős and Graham (Erdős and Graham, 1980) asked if there exists an n such that the divisors of n greater than 1 are the moduli of a distinct covering system with the following property: If there exists an integer which satisfies two congruences in the system, amodd and a′modd′, then gcd(d,d′)=1. We show that such an n does not exist. This problem is part of Problem # 204 on the website www.erdosproblems.com, compiled and maintained by Thomas Bloom.

A permutation of the integers avoiding monotone arithmetic progressions of length 6 was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length 5. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length 4. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length 5. A permutation of the positive integers that avoided monotone arithmetic progressions of length 4 with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length 4 with common difference not divisible by $2^k$. In addition, we specify the structure of permutations of $[1,n]$ that avoid length 3 monotone arithmetic progressions mod n as defined in (Davis et al. 1977) and provide an explicit construction for a multiplicative result on permutations that avoid length $k$ monotone arithmetic progressions mod n.