Universality for graphs with bounded density

Journal of Combinatorial Theory Series B · 2026-01-28

Extended VC-dimension, and Radon and Tverberg type theorems for unions of convex sets

Society for Industrial and Applied Mathematics eBooks · 2026-01-01

We define and study an extension of the notion of the VC-dimension of a hypergraph and apply it to establish a Tverberg type theorem for unions of convex sets. We also prove a new Radon type theorem for unions of convex sets and settle a well-known open problem posed by Kalai in the 1970s.

Random Cayley graphs and random sumsets

ArXiv.org · 2025-09-02

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in the sumset $A+A$ and is not much smaller than it. Using this result we obtain improved bounds for the problem of estimating the typical independence number of sparse random Cayley or Cayley-sum graphs, and for the problem of estimating the smallest size of a subset of $G$ which is not a sumset. We also obtain tight bounds for the typical maximum length of an arithmetic progression in the sumset of a sparse random subset of $G$.

Hitting k primes by dice rolls

ArXiv.org · 2025-02-12

Let $S=(d_1,d_2,d_3, \ldots )$ be an infinite sequence of rolls of independent fair dice. For an integer $k \geq 1$, let $L_k=L_k(S)$ be the smallest $i$ so that there are $k$ integers $j \leq i$ for which $\sum_{t=1}^j d_t$ is a prime. Therefore, $L_k$ is the random variable whose value is the number of dice rolls required until the accumulated sum equals a prime $k$ times. It is known that the expected value of $L_1$ is close to $2.43$. Here we show that for large $k$, the expected value of $L_k$ is $(1+o(1)) k\log_e k$, where the $o(1)$-term tends to zero as $k$ tends to infinity. We also include some computational results about the distribution of $L_k$ for $k \leq 100$.

Rainbow stackings of random edge‐colorings

Bulletin of the London Mathematical Society · 2025-03-13

Abstract A rainbow stacking of ‐edge‐colorings of the complete graph on vertices is a way of superimposing so that no edges of the same color are superimposed on each other. We determine a sharp threshold for (as a function of and ) governing the existence and nonexistence of rainbow stackings of random ‐edge‐colorings .

Ordering Candidates via Vantage Points

COMBINATORICA · 2025-04-01

The spanning tree spectrum: improved bounds and simple proofs

ArXiv.org · 2025-03-31

The number of spanning trees of a graph $G$, denoted $τ(G)$, is a well studied graph parameter with numerous connections to other areas of mathematics. In a recent remarkable paper, answering a question of Sedláček from 1969, Chan, Kontorovich and Pak showed that $τ(G)$ takes at least $1.1103^n$ different values across simple (and planar) $n$-vertex graphs $G$, for large enough $n$. We give a very short, purely combinatorial proof that at least $1.55^n$ values are attained. We also prove that exponential growth can be achieved with regular graphs, determining the growth rate in another problem first raised by Sedláček in the late 1960's. We further show that the following modular dual version of the result holds. For any integer $N$ and any $u < N$ there exists a planar graph on $O(\log N)$ vertices whose number of spanning trees is $u$ modulo $N$.

Distinct Directions and Distinct Distances in $\mathbb{R}^d$

ArXiv.org · 2025-08-12

We show that there exists an absolute positive constant $b (\geq \frac{1}{48})$ so that any set of $n$ points in $\mathbb{R}^d$ that is $d$-dimensional determines at least $bdn$ lines with pairwise distinct directions. As a consequence we prove that there are $d$-dimensional real norms $\|\cdot\|$ so that every set of $n>n_0(d)$ points that is $d$-dimensional determines at least $(bd-o(1))n$ distinct distances with respect to $\|\cdot \|$.

Induced matching treewidth and tree-independence number, revisited

ArXiv.org · 2025-11-05

We study two graph parameters defined via tree decompositions: tree-independence number and induced matching treewidth. Both parameters are defined similarly as treewidth, but with respect to different measures of a tree decomposition $\mathcal{T}$ of a graph $G$: for tree-independence number, the measure is the maximum size of an independent set in $G$ included in some bag of $\mathcal{T}$, while for the induced matching treewidth, the measure is the maximum size of an induced matching in $G$ such that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the matching. While the induced matching treewidth of any graph is bounded from above by its tree-independence number, the family of complete bipartite graphs shows that small induced matching treewidth does not imply small tree-independence number. On the other hand, Abrishami, Briański, Czyżewska, McCarty, Milanič, Rzążewski, and Walczak~[SIAM Journal on Discrete Mathematics, 2025] showed that, if a fixed biclique $K_{t,t}$ is excluded as an induced subgraph, then the tree-independence number is bounded from above by some function of the induced matching treewidth. The function resulting from their proof is exponential even for fixed $t$, as it relies on multiple applications of Ramsey's theorem. In this note we show, using the Kövári-Sós-Turán theorem, that for any class of $K_{t,t}$-free graphs, the two parameters are in fact polynomially related.

Essentially tight bounds for rainbow cycles in proper edge‐colourings

Proceedings of the London Mathematical Society · 2025-04-01 · 1 citations

Abstract An edge‐coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge‐coloured graph on vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of for this question. Very recently, Janzer–Sudakov and Kim–Lee–Liu–Tran independently removed the term in Tomon's bound, showing a bound of . We prove an upper bound of for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the term, and so, it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non‐abelian groups.