arXiv subjects

Israel R. Curbelo

Publications and source records attributed to Israel R. Curbelo.

7 recordsLinked to original sources

arXiv2026

Asymptotic Bounds for Online Ramsey Numbers of Stars versus Long Paths and Cycles

The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. In each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer $t$ for which Builder has a strategy guaranteeing a red copy of $G$ or a blue copy of $H$ within $t$ rounds. For every fixed integer $k\ge4$, the best-known lower bounds for $\tilde{r}(K_{1,k},P_n)$ and $\tilde{r}(K_{1,k},C_n)$ are $\left(\frac{k+3}{4}+o(1)\right)n$ as $n\to\infty$. We improve the corresponding asymptotic upper bounds from $(k+o(1))n$ to $\left(\frac{2k+4}{5}+o(1)\right)n$ as $n\to\infty$.

math.CO↗

arXiv2026

Online coloring of short interval graphs and two-count interval graphs

We study the online coloring of $σ$-interval graphs, which are interval graphs with interval lengths in $[1,σ]$ and 2-count interval graphs, which are interval graphs that require at most two distinct interval lengths. For $σ$-interval graphs, the Kierstead-Trotter algorithm has competitive ratio 3 and no online algorithm has competitive ratio better than 2. In this paper, we show that for every $\varepsilon>0$, there is a $σ>1$ such that there is no online algorithm for $σ$-interval coloring with competitive ratio less than $3-\varepsilon$. For 2-count interval graphs, we show that the greedy algorithm First-Fit has competitive ratio at most $4$, that there is no online algorithm with competitive ratio less than $2.5$ when the interval representation is unknown, and that there is no online algorithm with competitive ratio less than $2$ when the interval representation is known.

cs.DS↗

arXiv2026

On the asymptotic behavior of online Ramsey numbers for stars, paths and cycles

The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. Each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer $t$ for which Builder has a strategy that guarantees a red copy of $G$ or a blue copy of $H$ in at most $t$ rounds. We show that for every fixed $k$, there are constants $λ_1$ and $λ_2$ such that $\tilde{r}(P_k,P_n)/n$ and $\tilde{r}(P_k,C_n)/n$ converge to $λ_1$, and $\tilde{r}(K_{1,k},P_n)/n$ and $\tilde{r}(K_{1,k},C_n)/n$ converge to $λ_2$.

math.CO↗

arXiv2025

On the on-line coloring of unit interval graphs with proper interval representation

We define the problem as a two-player game between Algorithm and Builder. The game is played in rounds. Each round, Builder presents an interval that is neither contained in nor contains any previously presented interval. Algorithm immediately and irrevocably assigns the interval a color that has not been assigned to any interval intersecting it. The set of intervals form an interval representation for a unit interval graph and the colors form a proper coloring of that graph. For every positive integer $ω$, we define the value $R(ω)$ as the maximum number of colors for which Builder has a strategy that forces Algorithm to use $R(ω)$ colors with the restriction that the unit interval graph constructed cannot contain a clique of size $ω+1$. In 1981, Chrobak and Ślusarek showed that $R(ω)\leq2ω-1$. In 2005, Epstein and Levy showed that $R(ω)\geq\lfloor{3ω/2\rfloor}$. This problem remained unsolved for $ω\geq 3$. In 2023, Biró and Curbelo showed that $R(3)=5$. In this paper, we show that $R(4)=7$

math.CO↗

arXiv2023

Improved lower bounds on the on-line chain partitioning of posets of bounded dimension

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemerédi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In a survey paper by Bosek et al., it is shown that Szemerédi's argument could be improved to obtain a lower bound almost twice as good. Variants of the problem were considered where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size $d$. In this paper, we prove two results. First, we prove that any on-line algorithm can be forced to use $(2-o(1))\binom{w+1}{2}$ chains to partition a $2$-dimensional poset of width $w$. Second, we prove that any on-line algorithm can be forced to use $(2-\frac{1}{d-1}-o(1))\binom{w+1}{2}$ chains to partition a poset of width $w$ presented via a realizer of size $d$.

math.CO↗

arXiv2022

Improved lower bound on the on-line chain partitioning of semi-orders with representation

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains in the partition. The on-line chain partitioning problem involves finding the minimal number of chains needed by an optimal on-line algorithm. Chrobak and Ślusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most $3w-2$ chains, where $w$ is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most $2w-1$ chains, over $30$ years later, it remains unknown whether a more optimal algorithm exists. In this paper, we improve upon previously known bounds and show that any on-line algorithm can be forced to use $\lceil\frac{3}{2}w\rceil$ chains to partition a semi-order presented in the form of its unit-interval representation. As a consequence, we completely solve the problem for $w=3$.

math.CO↗

arXiv2021

Weak independence of events and the converse of the Borel--Cantelli Lemma

The converse of the Borel-Cantelli Lemma states that if $\{A_i\}_{i=1}^\infty$ is a sequence of independent events such that $\sum P(A_i)=\infty$, then almost surely infinitely many of these events will occur. Erd\H os and Rényi proved that it is sufficient to weaken the condition of independence to pairwise independence. In this paper we study various conditions of weak independence that imply the conclusion of the converse of the Borel--Cantelli Lemma. We will determine the exact implicational relationship among these conditions.

math.PR↗