Problems in extremal and probabilistic combinatorics: cubes, squares and permutations
Keywords: zero-sum
lazy transpositions
transposition shuffles
reachability
permutation patterns
square-free
hypercubes
intersection sizes
twins in permutations
boolean functions

We begin by studying the possible intersection sizes of a k-dimensional linear subspace with the hypercube {0,1}n. For a fixed k, the largest intersection size is 2k and it was shown by Melo and Winter that the second largest intersection size is 2k−1+2k−2. We show that all intersections sizes larger than 2k−1 are of the form 2k−1+2i or 35⋅2k−6, completely determining the large'' intersection sizes and disproving a conjecture of Melo and Winter. We also show that the only possible intersection size in the interval (15⋅2k−5,2k−1) is 63⋅2k−7, disproving a second conjecture of Melo and Winter that all of the small'' values are present.
Next we consider Lipschitz bijections between functions from {0,1}n to {0,1}n and consider how much these bijections must stretch'' the edges of the hypercube. We answer four questions posed by Rao and Shinkar.
1. We show that there is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(1)-input bits.
2. We give a construction for a mapping from XOR to Majority which has average stretch O(
√
n
), matching a previously known lower bound.
3. We give a 3-Lipschitz embedding ϕ:{0,1}n→{0,1}2n+1 such that XOR(x)=Majority(ϕ(x)) for all x∈{0,1}n.
4. We show that with high probability there is a O(1)-bi-Lipschitz mapping from Dictator to a uniformly random balanced function.

Moving from (hyper)cubes to squares, we consider a problem about zero-sum squares in {−1,1}-matrices. Given a matrix M=(ai,j) a {square} is a 2×2 submatrix with entries ai,j, ai,j+s, ai+s,j, ai+s,j+s for some s≥1, and a {zero-sum square} is a square where the entries sum to 0. Recently, Ar\'evalo, Montejano and Rold\'an-Pensado proved that all large n×n {−1,1}-matrices with discrepancy |∑ai,j|≤n contain a zero-sum square unless they are split, and conjectured that this can be improved to Cn for large enough n. We show that a quadratic bound holds: all large n×n {−1,1}-matrices with discrepancy at most n2/4 are either split or contain a zero-sum square.
We next consider another problem regarding squares, but this time squares in permutations. A square in a permutation is a factor S=(S1;S2) where S1 and S2 have the same pattern, and a permutation is said to be square-free if it contains no non-trivial squares. The permutation is further said to be bicrucial with respect to squares if every extension to both the left and right contains a square. We completely classify for which n there exists a bicrucial square-free permutation, confirming two conjectures of Gent, Kitaev, Konovalov, Linton and Nightingale.
Two permutations σ and π are said to be ℓ-similar if they can be partitioned into subpermutations σ(1),…,σ(ℓ) and π(1),…,π(ℓ) respectively such that σ(i) and π(i) have the same pattern for all i∈[ℓ]. Clearly, two permutations may only be n-similar, but what happens in the average'' case? It is easy to get an upper bound of O(n1/2) using the Erd\H{o}s-Szekeres Theorem, and a lower bound of Ω(n1/3) follows from the length of the longest twin in a random permutation. We prove that two independent uniformly random permutations are O(n1/3log11/6(n))-similar with high probability, matching the lower bound up to a polylogarithmic factor.
Finally, we consider creating uniformly random permutations using sequences of independent lazy transpositions and ask how long such a sequence must be. We give an improved construction which uses (2/3)\binom{n}{2}+O(n\log(n)) lazy transpositions, giving a negative answer to a problem of Angel and Holroyd. The best-known lower bound of \Omega(n\log(n)) is based on the number of transpositions needed to reach'' each configuration with positive probability. We introduce similar reachability questions for smaller subsets and settle the case of subsets of size 2 exactly: the smallest sequence of transpositions such that each pair of distinct positions (x,y) has a subsequence mapping (1,2) to (x,y) has length \lceil 3n/2\rceil-2 . We also consider the case where all transpositions are of the form (1, \cdot) and use the structure in this case to provide the first separation between uniformity and reachability.