arXiv subjects

Luis Boza

Publications and source records attributed to Luis Boza.

3 recordsLinked to original sources

arXiv2026

New Upper Bounds for the Classical Ramsey Numbers $R(4,4,4)$, $R(3,4,5)$ and $R(3,3,6)$

The inequality \[ R(k_1,\ldots,k_r)\le 2-r+\sum_{i=1}^r R(k_1,\ldots,k_{i-1},k_i-1,k_{i+1},\ldots,k_r) \] is well known, and it is strict whenever the right-hand side and at least one of the terms in the sum are even. Except for two known cases, the best upper bounds for classical Ramsey numbers with at least three colors have so far been obtained from this inequality. In this paper we present new bounds such as $R(4,4,4)\le 229$, $R(3,4,5)\le 157$ and $R(3,3,6)\le 91$.

math.CO↗

arXiv2024

Exact Values and Bounds for Ramsey Numbers of $C_4$ Versus a Star Graph

We study Ramsey numbers of the form $R(C_4,K_{1,n})$. We determine the eight previously unknown values of $R(C_4,K_{1,n})$ for $n\le 38$. In particular, we show that $R(C_4,K_{1,27})=33$ and $R(C_4,K_{1,n})=n+7$ for $28\le n\le 33$ and for $n=37$. We also establish new general inequalities relating different values of this function. Specifically, if $m\equiv 2\pmod 6$ with $m\ge 8$, then $R(C_4,K_{1,m^2+3})\le m^2+m+4$, and for all positive integers $a$ and $b$, either $R(C_4,K_{1,a})\ge a+b$ or $R(C_4,K_{1,a+b})\le a+2b$. As consequences, we obtain the functional inequalities $f(2n-f(n)+1)\ge n$ and $f(f(n)+1)\le 2f(n)-n+2$, where $f(n)=R(C_4,K_{1,n})$.

math.CO↗

arXiv2023

Some Upper Bounds on Ramsey Numbers Involving $C_4$

We obtain some new upper bounds on the Ramsey numbers of the form $R(\underbrace{C_4,\ldots,C_4}_m,G_1,\ldots,G_n)$, where $m\ge 1$ and $G_1,\ldots,G_n$ are arbitrary graphs. We focus on the cases of $G_i$'s being complete, star $K_{1,k}$ or book graphs $B_k$, where $B_k=K_2+kK_1$. If $k\ge 2$, then our main upper bound theorem implies that $$R(C_4,B_k) \le R(C_4,K_{1,k})+\left\lceil\sqrt{R(C_4,K_{1,k})}\right\rceil+1.$$ Our techniques are used to obtain new upper bounds in several concrete cases, including: $R(C_4,K_{11})\leq 43$, $R(C_4,K_{12})\leq 51$, $R(C_4,K_3,K_4)\leq 29$, $R(C_4, K_4,K_4)\leq 66$, $R(C_4,K_3,K_3,K_3)\leq 57$, $R(C_4,C_4,K_3,K_4)\leq 75$, and $R(C_4,C_4,K_4,K_4)\leq 177$, and also $R(C_4,B_{17})\leq 28$.

math.CO↗