Search

A natural open problem in Ramsey theory is to determine those $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H, K_n^{(3)})$ grows polynomially with $n$. We make substantial progress on this question by showing that if $H$ is tightly connected or has at most two tight components, then $r(H, K_n^{(3)})$ grows polynomially if and only if $H$ is not contained in an iterated blowup of an edge., Comment: 12 pages

A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{\Omega_H(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear., Comment: 22 pages

We study how many copies of a graph $F$ that another graph $G$ with a given number of cliques is guaranteed to have. For example, one of our main results states that for all $t\ge 2$, if $G$ is an $n$ vertex graph with $kn^{3/2}$ triangles and $k$ is sufficiently large in terms of $t$, then $G$ contains at least \[\Omega(\min\{k^t n^{3/2},k^{\frac{2t^2}{3t-1}}n^{\frac{5t-2}{3t-1}}\})\] copies of $K_{2,t}$, and furthermore, we show these bounds are essentially best-possible provided either $k\ge n^{1/2t}$ or if certain bipartite-analogues of well known conjectures for Tur\'an numbers hold., Comment: 21 pages

A family of sets $A$ is said to be an antichain if $x\not\subset y$ for all distinct $x,y\in A$, and it is said to be a distance-$r$ code if every pair of distinct elements of $A$ has Hamming distance at least $r$. Here, we prove that if $A\subset 2^{[n]}$ is both an antichain and a distance-$(2r+1)$ code, then $|A| = O_r(2^n n^{-r-1/2})$. This result, which is best-possible up to the implied constant, is a purely combinatorial strengthening of a number of results in Littlewood--Offord theory; for example, our result gives a short combinatorial proof of H\'alasz's theorem, while all previously known proofs of this result are Fourier-analytic., Comment: 8 pages

A group of players are supposed to follow a prescribed profile of strategies. If they follow this profile, they will reach a given target. We show that if the target is not reached because some player deviates, then an outside observer can identify the deviator. We also construct identification methods in two nontrivial cases., Comment: 21 pages, accepted journal version

We elucidate the relationship between the threshold and the expectation-threshold of a down-set. Qualitatively, our main result demonstrates that there exist down-sets with polynomial gaps between their thresholds and expectation-thresholds; in particular, the logarithmic gap predictions of Kahn--Kalai and Talagrand (recently proved by Park--Pham and Frankston--Kahn--Narayanan--Park) about up-sets do not apply to down-sets. Quantitatively, we show that any collection $\mathcal{G}$ of graphs on $[n]$ that covers the family of all triangle-free graphs on $[n]$ satisfies the inequality $\sum_{G \in \mathcal{G}} \exp(-\delta e(G^c) / \sqrt{n}) < 1/2$ for some universal $\delta > 0$, and this is essentially best-possible., Comment: 17 pages

What is the probability that a sparse $n$-vertex random $d$-regular graph $G_n^d$, $n^{1-c}

We consider the problem of bounding the number of permutations $\sigma\in S_n$ that avoid a fixed permutation $\pi\in S_k$ in specific indices given by a $k$-uniform hypergraph $\Lambda$. We obtain relatively sharp bounds in the case where $\Lambda$ is a random hypergraph, and find bounds in the case where $\Lambda$ contains many large cliques. Along the way, we prove a supersaturation version of F\"uredi-Hajnal, which may be of independent interest., Comment: 33 pages. v2, some restructuring and edits made for clarity. Added a discussion about concentration results

We consider asymptotics of set partition pattern avoidance in the sense of Klazar. Our main result derives the asymptotics of the number of set partitions avoiding a given set partition within an exponential factor, which leads to a classification of possible growth rates of set partition pattern classes. We further define a notion of permutation-tuple avoidance, which generalizes notions of Aldred et al. and the usual permutation pattern setting, and similarly determine the number of permutation-tuples avoiding a given tuple to within an exponential factor., Comment: 21 pages, 1 figure. Supersedes arxiv:1609.06023

We consider asymptotics of set partition pattern avoidance in the sense of Klazar. One of the results of this paper extends work of Alweiss, and finds a classification for set partitions $\pi$ such that the number of set partitions of $[n]$ avoiding $\pi$ grows more slowly than $n^{cn}$ for all $c>0$. Several conjectures are proposed, and the related question of asymptotics of parallel ($k$-tuple) permutation pattern avoidance is considered and solved completely to within an exponential factor, generalizing Marcus and Tardos's 2004 proof of the Stanley-Wilf Conjecture., Comment: arXiv:1707.07809 functions as a replacement for this paper. It was not submitted as a replacement as there is a second author on it

For $X$ a curve over a field of positive characteristic, we investigate when the canonical representation of $\text{Aut}(X)$ on $H^0(X, \Omega_X)$ is irreducible. Any curve with an irreducible canonical representation must either be superspecial or ordinary. Having a small automorphism group is an obstruction to having irreducible canonical representation; with this motivation, the bulk of the paper is spent bounding the size of automorphism groups of superspecial and ordinary curves. After proving that all automorphisms of an $\mathbb{F}_{q^2}$-maximal curve are defined over $\mathbb{F}_{q^2}$, we find all superspecial curves with $g > 82$ having an irreducible representation. In the ordinary case, we provide a bound on the size of the automorphism group of an ordinary curve that improves on a result of Nakajima., Comment: 29 pages

It is well known that for any prime $p\equiv 3$ (mod $4$), the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{-p})$, $h(p)$ and $h(-p)$ respectively, are odd. It is natural to ask whether there is a formula for $h(p)/h(-p)$ modulo powers of $2$. We show the formula $h(p) \equiv h(-p) m(p)$ (mod $16$), where $m(p)$ is an integer defined using the "negative" continued fraction expansion of $\sqrt{p}$. Our result solves a conjecture of Richard Guy., Comment: 9 pages; additional background given in introduction concerning $h(p)$ and $h(-p)$ modulo small powers of 2