arXiv subjects

Eion Mulrenin

Publications and source records attributed to Eion Mulrenin.

7 recordsLinked to original sources

arXiv2026

Some remarks on Folkman graphs for triangles

Folkman's theorem asserts the existence of graphs $G$ which are $K_4$-free, but which have the property that every two-coloring of $E(G)$ contains a monochromatic triangle. The quantitative aspects of $f(2,3,4)$, the least $n$ such that there exists an $n$-vertex graph with both properties above, are notoriously difficult; a series of improvements over the span of two decades witnessed the solution to two \$100 Erdős problems, and the current record due to Lange, Radziszowski, and Xu now stands at $f(2,3,4) \leq 786$,with another \$100 problem of Graham asking for a proof that $f(2,3,4) < 100$. In this paper, we study Folkman-like properties of a sequence $H_q$ of finite geometric graphs constructed using Hermitian unitals in projective planes and present some evidence that the graph $H_3$, which has 63 vertices, might contain a Folkman graph as a proper subgraph. More precisely, we first prove that for all prime powers $q \geq 3$, there exists a system $\mathscr{T}_q$ of triangles in $H_q$ such that no four span a $K_4$ in $H_q$, but every two-coloring of $E(H_q)$ induces a monochromatic triangle in $\mathscr{T}_q$. We then show that a certain random alteration of $H_q$ which destroys all of its $K_4$'s will, for large $q$, maintain the Ramsey property with high probability.

math.CO↗

arXiv2026

Recursive upper bounds for the vertex online Ramsey game with applications to hypergraph Ramsey numbers

The classical recursive upper bound on hypergraph Ramsey numbers due to Erdős and Rado states that for $2 \leq k < s \leq t$, \[ r_k(s,t) \leq 2^{\binom{r_{k-1}(s-1,t-1)}{k-1}}. \] In 2010, Conlon, Fox, and Sudakov introduced the so-called vertex online Ramsey numbers $\tilde{r}(s,t)$ for graphs to obtain a quantitative improvement over this bound when $k=3$. In this note, we show that the natural hypergraph generalization $\tilde{r}_k(s,t)$ of the vertex online Ramsey numbers satisfy an improved recurrence \[ \tilde{r}_k(s,t) \leq 2^{(1+o(1))\tilde{r}_{k-1}(s-1,t-1)}. \] We obtain several corollaries from this, including a lower-order improvement to the best known quantitative upper bounds for hypergraph Ramsey numbers and an improvement to the above recursive bound of Erdős and Rado.

math.CO↗

arXiv2026

Improved Ramsey bounds for generalized Schur equations

We show that for $m, r \in \mathbb{N}$ and $N > (2m+1)^r (r!)^{1/m}$, every $r$-coloring of the integers in the interval $[N]$ contains a monochromatic solution to the equation \[ x_1 + \dots + \dots x_{m+1} = y_1 + \dots + y_m. \] This generalizes and improves recent results of Koścuiszko. We also show that if $N \geq 2^{r}$, then every $r$-coloring of the integers in $[N]$ must always determine a monochromatic solution to the above equation for some $m \geq 1$. The latter estimate is optimal.

math.CO↗

arXiv2026

Two counterexamples to a conjecture about even cycles

A conjecture of Verstraëte states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for $\ell = 4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger's construction for extremal $C_{10}$-free graphs.

math.CO↗

arXiv2025

Color avoidance for monotone paths

In 2014, Moshkovitz and Shapira determined the tower height for hypergraph Ramsey numbers of tight monotone paths. We address the color-avoiding version of this problem in which one no longer necessarily seeks a monochromatic subgraph, but rather one which avoids some colors. This problem was previously studied in uniformity two by Loh and by Gowers and Long. We show, in general, that the tower height for such Ramsey numbers requires one less exponential than in the usual setting. The transition occurs at uniformity three, where the usual Ramsey numbers of monotone paths of length $n$ are exponential in $n$, but the color-avoiding Ramsey numbers turn out to be polynomial.

math.CO↗

arXiv2025

Even cycles in graphs avoiding longer even cycles

A conjecture of Verstraëte states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by Kühn and Osthus. We show that $C_6$ and $C_{2k}$ satisfy the conjecture for all odd $k$, but observe that a recent construction of a dense $C_{10}$-free subgraph of the hypercube yields a counterexample to the conjecture for $C_8$ and $C_{10}$.

math.CO↗

arXiv2024

Sharp exponents for bipartite Erdős-Rado numbers

The Erdős-Rado canonization theorem generalizes Ramsey's theorem to edge-colorings with an unbounded number of colors, in the sense that for $n = ER(m)$ sufficiently large, any edge-coloring of $E(K_n) \to \mathbb{N}$ will yield some copy of $K_m$ which is colored according to one of four canonical patterns. In this paper, we show that in the bipartite setting, the bipartite Erdős-Rado number $ER_B(m)$ satisfies \[ \log ER_B(m) = Θ(m \log m). \] Comparing this to the non-bipartite setting, the best known lower and upper bounds on $\log ER(m)$ are still separated by a factor of $\log m$.

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