arXiv subjects

Tom Bohman

Publications and source records attributed to Tom Bohman.

At least 19 recordsLinked to original sources

arXiv2026

The largest $K_r$-free set of vertices in a random graph

For $r \ge 2$ and a graph $G$, let $\alpha_{{r}}(G)$ be the maximum number of vertices in a $K_r$-free subgraph of $G$. We investigate the value $\alpha_{r}(G)$ when $G$ is the random graph $G \sim G_{n, 1/2}$ and discover the following phenomenon: with high probability, $\alpha_r(G)$ lies in an interval of constant length that varies in a non-monotonic fashion from $1$ to $\lfloor r/2\rfloor+1$ depending on the value of $n$. The special case $r=2$ corresponds to the independence number of random graphs which is well-known to have two-point concentration; our results therefore extend and generalize this basic fact in random graph theory, showing more complicated behavior when $r>2$. We also prove similar results where $K_r$ is replaced by any color critical graph like $C_5$.

math.CO↗

arXiv2024

A note on Two-Point Concentration of the Independence Number of $G_{n,m}$

We show that the independence number of $ G_{n,m}$ is concentrated on two values for $ n^{5/4+ \epsilon} < m \le \binom{n}{2}$. This result establishes a distinction between $G_{n,m}$ and $G_{n,p}$ with $p = m/ \binom{n}{2}$ in the regime $ n^{5/4 + \epsilon} < m< n^{4/3}$. In this regime the independence number of $ G_{n,m}$ is concentrated on two values while the independence number of $ G_{n,p}$ is not; indeed, for $p$ in this regime variations in $ \alpha( G_{n,p})$ are determined by variations in the number of edges in $ G_{n,p}$.

math.CO↗

arXiv2024

Two-Point Concentration of the Domination Number of Random Graphs

We show that the domination number of the binomial random graph G_{n,p} with edge-probability p is concentrated on two values for p \ge n^{-2/3+\eps}, and not concentrated on two values for general p \le n^{-2/3}. This refutes a conjecture of Glebov, Liebenau and Szabo, who showed two-point concentration for p \ge n^{-1/2+\eps}, and conjectured that two-point concentration fails for p \ll n^{-1/2}. The proof of our main result requires a Poisson type approximation for the probability that a random bipartite graph has no isolated vertices, in a regime where standard tools are unavailable (as the expected number of isolated vertices is relatively large). We achieve this approximation by adapting the proof of Janson's inequality to this situation, and this adaptation may be of broader interest.

math.PR↗

arXiv2022

Two-Point Concentration of the Independence Number of the Random Graph

We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ \epsilon} < p \le 1$. This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in general, concentrated on 2 values for $ p = o \left( (\log(n)/n)^{2/3} \right)$. The extent of concentration of the independence number of $ G_{n,p}$ for $ \omega(1/n) <p \le n^{-2/3}$ remains an interesting open question.

math.CO↗

arXiv2022

A Critical Probability for Biclique Partition of $G_{n,p}$

The biclique partition number of a graph $G= (V,E)$, denoted $bp(G)$, is the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that $ bp(G) \leq n - \alpha(G)$, where $\alpha(G)$ is the maximum size of an independent set of $G$. Erd\H{o}s conjectured in the 80's that for almost every graph $G$ equality holds; i.e., if $ G=G_{n,1/2}$ then $bp(G) = n - \alpha(G)$ with high probability. Alon showed that this is false. We show that the conjecture of Erd\H{o}s is true if we instead take $ G=G_{n,p}$, where $p$ is constant and less than a certain threshold value $p_0 \approx 0.312$. This verifies a conjecture of Chung and Peng for these values of $p$. We also show that if $p_0 < p <1/2$ then $bp(G_{n,p}) = n - (1 + \Theta(1)) \alpha(G_{n,p})$ with high probability.

math.CO↗

arXiv2022

Complexes of nearly maximum diameter

The diameter of a strongly connected $d$-dimensional simplicial complex is the diameter of its dual graph. We provide a probabilistic proof of the existence of $d$-dimensional simplicial complexes with diameter $ (\frac{1}{d \cdot d!} - (\log n)^{-\epsilon}) n^d$. Up to the first order term, this is the best possible lower bound for the maximum diameter of a $d$-complex on $n$ vertices as a simple volume argument shows that the diameter of a $d$-dimensional simplicial complex is at most $ \frac{1}{d} \binom{n}{d}$. We also find the right first-order asymptotics for the maximum diameter of a $d$-pseudomanifold on $n$ vertices.

math.CO↗

arXiv2021

Coprime Mappings and Lonely Runners

For $x$ real, let $ \{ x \}$ be the fractional part of $x$ (i.e. $\{x\} = x - \lfloor x \rfloor $). The lonely runner conjecture can be stated as follows: for any $n$ positive integers $ v_1 < v_2 < \dots < v_n $ there exists a real number $t$ such that $ 1/(n+1) \le \{ v_i t\} \le n/(n+1) $ for $ i = 1, \dots, n$. In this paper we prove that if $ ε>0 $ and $n$ is sufficiently large (relative to $ε$) then such a $t$ exists for any collection of positive integers $ v_1 < v_2 < \dots < v_n$ such that $ v_n < (2-ε)n$. This is an approximate version of a natural next step for the study of the lonely runner conjecture suggested by Tao. The key ingredient in our proof is a result on coprime mappings. Let $A$ and $B$ be sets of integers. A bijection $ f:A \to B$ is a coprime mapping if $ a $ and $f(a)$ are coprime for every $ a \in A$. We show that if $A,B \subset [n]$ are intervals of length $2m$ where $ m = e^{ Ω({(\log\log n)}^2)}$ then there exists a coprime mapping from $A$ to $B$. We do not believe that this result is sharp.

math.NT↗

arXiv2021

A Construction for Boolean cube Ramsey numbers

Let $Q_n$ be the poset that consists of all subsets of a fixed $n$-element set, ordered by set inclusion. The poset cube Ramsey number $R(Q_n,Q_n)$ is defined as the least $m$ such that any 2-coloring of the elements of $Q_m$ admits a monochromatic copy of $Q_n$. The trivial lower bound $R(Q_n,Q_n)\ge 2n$ was improved by Cox and Stolee, who showed $R(Q_n,Q_n)\ge 2n+1$ for $3\le n\le 8$ and $n\ge 13$ using a probabilistic existence proof. In this paper, we provide an explicit construction that establishes $R(Q_n,Q_n)\ge 2n+1$ for all $n\ge 3$. The best known upper bound, due to Lu and Thompson, is $ R(Q_n, Q_n) \le n^2 - 2n + 2$.

math.CO↗

arXiv2021

Independent sets in hypergraphs omitting an intersection

A $k$-uniform hypergraph with $n$ vertices is an $(n,k,\ell)$-omitting system if it does not contain two edges whose intersection has size exactly $\ell$. If in addition it does not contain two edges whose intersection has size greater than $\ell$, then it is an $(n,k,\ell)$-system. Rödl and Šiňajová proved a lower bound for the independence number of $(n,k,\ell)$-systems that is sharp in order of magnitude for fixed $2 \le \ell \le k-1$. We consider the same question for the larger class of $(n,k,\ell)$-omitting systems. For $k\le 2\ell+1$, we believe that the behavior is similar to the case of $(n,k,\ell)$-systems and prove a nontrivial lower bound for the first open case $\ell=k-2$. For $k>2\ell+1$ we give new lower and upper bounds which show that the minimum independence number of $(n,k,\ell)$-omitting systems has a very different behavior than for $(n,k,\ell)$-systems. Our lower bound for $\ell=k-2$ uses some adaptations of the random greedy independent set algorithm, and our upper bounds (constructions) for $k> 2\ell+1$ are obtained from some pseudorandom graphs. We also prove some related results where we forbid more than two edges with a prescribed common intersection size and this leads to some applications in Ramsey theory. For example, we obtain good bounds for the Ramsey number $r_{k}(F^{k},t)$, where $F^{k}$ is the $k$-uniform Fan. Here the behavior is quite different than the case $k=2$ which reduces to the classical graph Ramsey number $r(3,t)$.

math.CO↗

arXiv2020

Hamilton cycles in 3-out

Let G_{\rm 3-out} denote the random graph on vertex set [n] in which each vertex chooses 3 neighbors uniformly at random. Note that G_{\rm 3-out} has minimum degree 3 and average degree 6. We prove that the probability that G_{\rm 3-out} is Hamiltonian goes to 1 as n tends to infinity.

math.CO↗

arXiv2019

Dynamic concentration of the triangle-free process

The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on the Ramsey numbers R(3,t): we show R(3,t) > (1-o(1)) t^2 / (4 log t), which is within a 4+o(1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self-correcting nature of key statistics of the process. Furthermore, we determine which bounded size subgraphs are likely to appear in the maximal triangle-free graph produced by the triangle-free process: they are precisely those triangle-free graphs with density at most 2.

math.CO↗

arXiv2019

On multicolor Ramsey numbers of triple system paths of length 3

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $ \mathcal{L}$ be the loose 3-uniform path with 3 edges and $ \mathcal{M}$ denote the messy 3-uniform path with 3 edges; that is, let $\mathcal{L} = \{abc, cde, efg\}$ and $\mathcal{M} = \{ abc, bcd, def\}$. In this note we prove $ r_k(\mathcal{L}) < 1.54k$ and $ r_k(\mathcal{M}) < 1.6k$ for $k$ sufficiently large.

math.CO↗

arXiv2019

Large girth approximate Steiner triple systems

In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at least g-2 triples.) We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6-o(1))n^2 triples and girth larger than \ell. The process iteratively adds random triples subject to the constraint that the girth remains larger than \ell. Our result is best possible up to the o(1)-term, which is a negative power of n.

math.CO↗

arXiv2018

A natural barrier in random greedy hypergraph matching

Let $r \ge 2$ be a fixed constant and let $ {\mathcal H}$ be an $r$-uniform, $D$-regular hypergraph on $N$ vertices. Assume further that $ D \to \infty$ as $N \to \infty$ and that degrees of pairs of vertices in ${\mathcal H}$ are at most $L$ where $L \ = D/ (\log N)^{ω(1)}$. We consider the random greedy algorithm for forming a matching in $ \mathcal{H}$. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of $ {\mathcal H}$ that are not saturated by the final matching is at most $ (L/D)^{ \frac{ 1}{ 2(r-1) } + o(1) } $. This point is a natural barrier in the analysis of the random greedy hypergraph matching process.

math.CO↗

arXiv2018

Independence number of graphs with a prescribed number of cliques

We consider the following problem posed by Erdos in 1962. Suppose that $G$ is an $n$-vertex graph where the number of $s$-cliques in $G$ is $t$. How small can the independence number of $G$ be? Our main result suggests that for fixed $s$, the smallest possible independence number undergoes a transition at $t=n^{s/2+o(1)}$. In the case of triangles ($s=3$) we obtain the following result which is sharp apart from constant factors and generalizes basic results in Ramsey theory: there exists $c>0$ such that every $n$-vertex graph with $t$ triangles has independence number at least $$c \cdot \min\left\{ \sqrt {n \log n}\, , \, \frac{n}{t^{1/3}} \left(\log \frac{n}{ t^{1/3}}\right)^{2/3} \right\}.$$

math.CO↗

arXiv2016

On randomly generated intersecting hypergraphs II

Let $c$ be a positive constant. Suppose that $r=o(n^{5/12})$ and the members of $\binom{[n]}{r}$ are chosen sequentially at random to form an intersecting hypergraph $\mathcal{H}$. We show that whp $\mathcal{H}$ consists of a simple hypergraph $\mathcal{S}$ of size $Θ(r/n^{1/3})$, a distinguished vertex $v$ and all $r$-sets which contain $v$ and meet every edge of $\mathcal{S}$. This is a continuation of the study of such random intersecting systems started in [Electron. J. Combin, (2003) R29] where the case $r=O(n^{1/3})$ was considered. To obtain the stated result we continue to investigate this question in the range $ω(n^{1/3})\le r \le o(n^{5/12})$.

math.CO↗

arXiv2016

On randomly generated intersecting hypergraphs

Let $c$ be a positive constant. We show that if $r=\lfloor cn^{1/3}\rfloor$ and the members of ${[n]\choose r}$ are chosen sequentially at random to form an intersecting hypergraph then with limiting probability $(1+c^3)^{-1}$, as $n\to\infty$, the resulting family will be of maximum size ${n-1\choose r-1}$.

math.CO↗

arXiv2016

How many random edges make a dense graph hamiltonian?

This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph hamiltonian with high probability. Adding $Θ(n)$ random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs.

math.CO↗