Erdős-Gyárfás problem for partially ordered sets

Abstract: Given integers $p,q,t$ with $1 \le t \le p$ and $1 \le q \le h_p(t)$, a strong $(p,q,t)$-coloring of the Boolean lattice $B_n$ is a coloring of its $t$-chains such that every induced copy of $B_p$ in $B_n$ uses at least $q$ colors on its $t$-chains. Let $f_t^{\sharp}(n,p,q)$ denote the minimum number of colors in such a coloring. We study this Boolean-lattice analogue of the Erdős-Gyárfás function. We first show that every finite poset strongly embeds into a Boolean lattice. Combined with a structural Ramsey theorem for finite posets with linear extensions, this implies the existence of the strong Boolean Ramsey number $\mathrm{R}^{\sharp}_{k,t}(\mathcal{B}\mid Q)$ for every integer $k\ge1$, every $t\ge1$, and every nonempty finite poset $Q$. In particular, this gives an affirmative answer to a problem of Cox and Stolee and yields the existence of $f_t^{\sharp}(n,p,2)$. Next, using the symmetric Lovász local lemma, we obtain a probabilistic upper bound on $f_t^{\sharp}(n,p,q)$. Finally, we prove lower bounds by combining Turán-type extremal estimates for $t$-chains, a double-counting argument, and a generalized Lubell-type framework for $t$-chains.

• The paper establishes a strong (p,q,t)-coloring for t-chains in Boolean lattices, adapting the classic Erdős-Gyárfás function to a poset context.
• It employs the Lovász Local Lemma to derive probabilistic upper bounds and supports these results with combinatorial lower bounds.
• The findings yield new lower bounds for strong Boolean Ramsey numbers, extending Ramsey theory to arbitrary finite posets and structured lattices.

Summary of "Erdős-Gyárfás problem for partially ordered sets" (2604.10229)

Introduction and Problem Framework

The paper addresses an adaptation of the classical Erdős-Gyárfás function—originally formulated for graph colorings—to the context of partially ordered sets (posets), specifically Boolean lattices. In the standard Erdős-Gyárfás problem, $f(n,p,q)$ is the minimum number of colors needed to edge-color the complete graph $K_n$ such that every $K_p$ subgraph uses at least $q$ distinct edge colors. This work introduces a natural analogue for the Boolean lattice $B_n = 2^{[n]}$, where $t$-chains (totally ordered subsets of size $t$) play the role of edges, and sublattices isomorphic to $B_p$ stand in for $K_p$ subgraphs.

The central object is the strong $(p,q,t)$-coloring: a coloring of the $K_n$0-chains of $K_n$1 so that every induced copy of $K_n$2 in $K_n$3 uses at least $K_n$4 colors. The function $K_n$5 denotes the minimum number of colors possible in such a coloring. This function generalizes classic notions in poset Ramsey theory and connects the extremal combinatorics of chains and antichains with Ramsey-type coloring constraints.

Existence and Ramsey-Theoretic Consequences

A fundamental contribution of the paper is to rigorously establish the existence of the strong Boolean Ramsey number $K_n$6 for every $K_n$7, $K_n$8, and nonempty finite poset $K_n$9. The authors first prove that every finite poset strongly embeds into a Boolean lattice (specifically, $K_p$0 embeds into $K_p$1 as an induced subposet), ensuring that Boolean lattices are sufficiently rich universes for poset Ramsey arguments.

Utilizing a structural Ramsey theorem for posets with linear extensions, it is shown that, for all such $K_p$2, the corresponding strong Ramsey number is finite. This positively resolves a problem posed by Cox and Stolee regarding the general existence of Ramsey numbers for induced poset substructures and establishes the existence of $K_p$3 for Boolean lattices in particular.

Monotonicity and Extremal Properties

Several monotonicity results are derived for the coloring function $K_p$4, mirroring properties known in classical Ramsey theory:

• Weak versus strong colorings: $K_p$5, since every strong embedding is also a weak embedding.
• Parameter monotonicity: $K_p$6 is non-increasing in $K_p$7 and non-increasing in $K_p$8 for fixed $K_p$9 and $q$0.
• Monotonicity in $q$1: $q$2 for $q$3, allowing for inductive bounds on Ramsey numbers.

These properties facilitate the inductive and probabilistic techniques developed in subsequent sections.

Probabilistic Upper Bounds via the Lovász Local Lemma

A cornerstone of the analysis is a probabilistic method application, leveraging the symmetric Lovász Local Lemma (LLL) to derive upper bounds for $q$4. The coloring is constructed by randomly assigning colors to each $q$5-chain independently. For a given induced $q$6 sublattice, the bad event is that its $q$7-chains use at most $q$8 colors. The LLL is invoked to show that, provided the number of colors is sufficiently large (relative to $q$9), the probability that no bad event occurs is positive, thereby guaranteeing the existence of a valid coloring.

The main result quantitatively asserts:

$B_n = 2^{[n]}$0

for some constant $B_n = 2^{[n]}$1 and all $B_n = 2^{[n]}$2, where $B_n = 2^{[n]}$3 is the number of $B_n = 2^{[n]}$4-chains in $B_n = 2^{[n]}$5 [see Theorem~\ref{th-upper-Pr} in the paper]. This bound tightly reflects the extremal growth in the number of $B_n = 2^{[n]}$6-chains and induced Boolean sublattices.

Combinatorial Lower Bounds

The authors complement their upper bounds with combinatorial lower bounds for $B_n = 2^{[n]}$7, based on:

• Turán-type extremal estimates for $B_n = 2^{[n]}$8-chains,
• Double-counting arguments involving induced copies and minimal $B_n = 2^{[n]}$9-chain incidences,
• Generalized Lubell-type functions that count the total number of $t$0-chains across color classes.

The main lower bound is:

$t$1

where $t$2 counts $t$3-chains in $t$4, $t$5 is the number of induced $t$6 copies in $t$7, $t$8 is the minimum number of induced $t$9 copies containing a given $t$0-chain, and the first term is often achievable for well-chosen parameters [see Theorem~\ref{th123}].

These results expose the fundamental tension between coloring constraints and induced subposet structure, yielding instances where the upper and lower bounds are provably asymptotic.

Applications to Boolean Ramsey Numbers

The probabilistic upper bound for $t$1 leads directly to new lower bounds for strong Boolean Ramsey numbers—parameters expressing the threshold dimension $t$2 so that any $t$3-coloring of $t$4-chains in $t$5 contains a monochromatic induced $t$6. Specifically, for large $t$7,

$t$8

improving upon earlier (non-logarithmic) bounds [Proposition~\ref{pro-compa}]. The authors also generalize the approach to arbitrary posets $t$9 and demonstrate that logarithmic lower bounds for the strong poset Ramsey numbers exist universally for finite posets, greatly expanding the reach of the classical Ramsey theory for posets.

Implications and Future Directions

This work strengthens the bridge between extremal combinatorics for chains and chains in posets and the landscape of Ramsey-theoretic coloring constraints. The combined application of probabilistic (LLL-based) upper bounds and combinatorial lower bounds provides a nearly tight understanding of the coloring thresholds in Boolean lattices for induced subposets. These results have further implications for:

• Understanding Ramsey phenomena in highly structured posets and lattices
• Generalizing to other host posets, such as grids and distributive lattices
• Clarifying the separation between weak versus strong coloring constraints
• Developing explicit constructions or algorithms for extremal colorings

Several open problems are noted regarding sharp determination of $B_p$0 values, the gap between weak and strong functions, and extensions to other poset families.

Conclusion

The paper systematically develops the theory of the Erdős-Gyárfás problem for Boolean lattices, introduces a Boolean-lattice version of the Erdős-Gyárfás function, and establishes its existence, structural properties, and tight probabilistic and combinatorial bounds. These advances deepen the connection between Ramsey theory, extremal poset theory, and combinatorial coloring, opening new avenues for research in combinatorics and discrete mathematics.