Uniform eventown problems

Let n≥. k. l≥. 2 be integers, and let F be a family of k-element subsets of an n-element set. Suppose that l divides the size of the intersection of any two (not necessarily distinct) members in F. We prove that the size of F is at most ([n/l][k/l]) provided n is sufficiently large for fixed k and l

Small doubling, atomic structure and ℓ\ell-divisible set families

Let $\mathcal{F}\subset 2^{[n]}$ be a set family such that the intersection of any two members of $\mathcal{F}$ has size divisible by $\ell$. The famous Eventown theorem states that if $\ell=2$ then $|\mathcal{F}|\leq 2^{\lfloor n/2\rfloor}$, and this bound can be achieved by, e.g., an `atomic' construction, i.e. splitting the ground set into disjoint pairs and taking their arbitrary unions. Similarly, splitting the ground set into disjoint sets of size $\ell$ gives a family with pairwise intersections divisible by $\ell$ and size $2^{\lfloor n/\ell\rfloor}$. Yet, as was shown by Frankl and Odlyzko, these families are far from maximal. For infinitely many $\ell$, they constructed families $\mathcal{F}$ as above of size $2^{\Omega(n\log \ell/\ell)}$. On the other hand, if the intersection of {\em any number} of sets in $\mathcal{F}\subset 2^{[n]}$ has size divisible by $\ell$, then it is easy to show that $|\mathcal{F}|\leq 2^{\lfloor n/\ell\rfloor}$. In 1983 Frankl and Odlyzko conjectured that $|\mathcal{F}|\leq 2^{(1+o(1)) n/\ell}$ holds already if one only requires that for some $k=k(\ell)$ any $k$ distinct members of $\mathcal{F}$ have an intersection of size divisible by $\ell$. We completely resolve this old conjecture in a strong form, showing that $|\mathcal{F}|\leq 2^{\lfloor n/\ell\rfloor}+O(1)$ if $k$ is chosen appropriately, and the $O(1)$ error term is not needed if (and only if) $\ell \, | \, n$, and $n$ is sufficiently large. Moreover the only extremal configurations have `atomic' structure as above. Our main tool, which might be of independent interest, is a structure theorem for set systems with small 'doubling'

The Space Complexity of Mirror Games

We consider the following game between two players Alice and Bob, which we call the mirror game. Alice and Bob take turns saying numbers belonging to the set {1, 2, ...,N}. A player loses if they repeat a number that has already been said. Otherwise, after N turns, when all the numbers have been spoken, both players win. When N is even, Bob, who goes second, has a very simple (and memoryless) strategy to avoid losing: whenever Alice says x, respond with N+1-x. The question is: does Alice have a similarly simple strategy to win that avoids remembering all the numbers said by Bob? The answer is no. We prove a linear lower bound on the space complexity of any deterministic winning strategy of Alice. Interestingly, this follows as a consequence of the Eventown-Oddtown theorem from extremal combinatorics. We additionally demonstrate a randomized strategy for Alice that wins with high probability that requires only O~(sqrt N) space (provided that Alice has access to a random matching on K_N). We also investigate lower bounds for a generalized mirror game where Alice and Bob alternate saying 1 number and b numbers each turn (respectively). When 1+b is a prime, our linear lower bounds continue to hold, but when 1+b is composite, we show that the existence of a o(N) space strategy for Bob (when N != 0 mod (1+b)) implies the existence of exponential-sized matching vector families over Z^N_{1+b}

The Slice Rank Polynomial Method

Suppose you wanted to bound the maximum size of a set in which every k-tuple of elements satisfied a specific condition. How would you go about this? Introduced in 2016 by Terence Tao, the slice rank polynomial method is a recently developed approach to solving problems in extremal combinatorics using linear algebraic tools. We provide the necessary background to understand this method, as well as some applications. Finally, we investigate a generalization of the slice rank, the partition rank introduced by Eric Naslund in 2020, along with various discussions on the intuition behind the slice rank polynomial method and other possible avenues for generalization

Independence numbers of Johnson-type graphs

We consider a family of distance graphs in $\mathbb{R}^n$ and find its independent numbers in some cases. Define graph $J_{\pm}(n,k,t)$ in the following way: the vertex set consists of all vectors from $\{-1,0,1\}^n$ with $k$ nonzero coordinates; edges connect the pairs of vertices with scalar product $t$. We find the independence number of $J_{\pm}(n,k,t)$ for $n > n_0 (k,t)$ in the cases $t = 0$ and $t = -1$; these cases for $k = 3$ are solved completely. Also the independence number is found for negative odd $t$ and $n > n_0 (k,t)$

Exact k -Wise Intersection Theorems

A typical problem in extremal combinatorics is the following. Given a large number n and a set L, find the maximum cardinality of a family of subsets of a ground set of n elements such that the intersection of any two subsets has cardinality in L. We investigate the generalization of this problem, where intersections of more than 2 subsets are considered. In particular, we prove that when k−1 is a power of 2, the size of the extremal k-wise oddtown family is (k−1)(n− 2log2(k−1)). Tight bounds are also found in several other basic case

HOMOGENEITY OF MAXIMAL ANTIPODAL SETS

We introduce a concept of connectedness of antipodal sets of compact Riemannian symmetric spaces and construct a method to make a bigger antipodal set from a given antipodal set. Moreover, using the connectedness we give a sufficient condition that a given maximal antipodal set is homogeneous

Invitation to intersection problems for finite sets

Extremal set theory is dealing with families, . F of subsets of an . n-element set. The usual problem is to determine or estimate the maximum possible size of . F, supposing that . F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any . r subsets in . F have at least . t elements in common, all the sizes of pairwise intersections belong to a fixed set, . L of natural numbers, there are no . s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bound