Universality of asymptotic graph homomorphism

TL;DR

This work proves that the asymptotic cohomomorphism order on finite graphs is countably universal, meaning every countable preorder embeds into it. The authors achieve this by embedding a known universal preorder on sets of binary strings into graphs using a construction based on lines and spectral points from the asymptotic spectrum, together with a new result equating the complement of the projective rank with the fractional clique covering number on fraction graphs to simulate rationals. They also develop a continuous family of fraction-normalised spectral points, yielding refined control over order embeddings, and provide warm-up universality results for simpler preorders. The findings illuminate the intrinsic complexity of Shannon capacity via the asymptotic spectrum and offer a new proof of non-asymptotic cohomomorphism universality, with potential extensions to Strassen semirings and related combinatorial frameworks.

Abstract

The Shannon capacity of graphs, introduced by Shannon in 1956 to model zero-error communication, asks for determining the rate of growth of independent sets in strong powers of graphs. Much is still unknown about this parameter, for instance whether it is computable. Recent work has established a dual characterization of the Shannon capacity in terms of the asymptotic spectrum of graphs. A core step in this duality theory is to shift focus from Shannon capacity itself to studying the asymptotic relations between graphs, that is, the asymptotic cohomomorphisms. Towards understanding the structure of Shannon capacity, we study the "combinatorial complexity" of asymptotic cohomomorphism. As our main result, we prove that the asymptotic cohomomorphism order is universal for all countable preorders. That is, we prove that any countable preorder can be order-embedded into the asymptotic cohomomorphism order (i.e. appears as a suborder). Previously this was only known for (non-asymptotic) cohomomorphism. Our proof is based on techniques from asymptotic spectrum duality and convex structure of the asymptotic spectrum of graphs. Our approach in fact leads to a new proof of the universality of (non-asymptotic) cohomomorphism.

Figures (7)

• Figure 1: Left: If $v <_W w$, then $\ell_v$ lies below $\ell_{w}$. Right: The blue lines $\ell_{w}$ for $v \nleq_W w$ have the property that at the witness value $r_v$, the line $\ell_v$ lies above all the blue lines.
• Figure 2: At first, we construct the line $\ell_\emptyset$. Any point $r_{\emptyset} \in (s,t)$ can be chosen as witness point.
• Figure 3: Lines $\ell_0, \ell_1$ are situated strictly below $\ell_\emptyset$ on $[s,t]$. For simplicity we let them intersect at $r_\emptyset$. We pick $r_1$ to the left of the intersection point $r_\emptyset$ so that $\ell_1(r_1) > \ell_0(r_1)$. Analogously, we choose $r_0$ to the right of the the intersection $r_{\emptyset}$ point so that $\ell_0(r_0) > \ell_1(r_0)$.
• Figure 4: We construct the lines $\ell_{01}$ and $\ell_{00}$ as slight "perturbations" of the line $\ell_0.$
• Figure 5: We construct lines $\ell_{10}$ and $\ell_{11}$ as slight "perturbations" of the line $\ell_1.$

Theorems & Definitions (44)

• Theorem 1
• Theorem 2: SpectrumGraphs
• Lemma 3: vrana2019probabilisticrefinementasymptoticspectrum
• Lemma 4
• Theorem 5
• Lemma 6: vrana2019probabilisticrefinementasymptoticspectrum, see also deboer2024asymptoticspectrumdistancegraph
• Lemma 7: vrana2019probabilisticrefinementasymptoticspectrum
• Lemma 8: vrana2019probabilisticrefinementasymptoticspectrum
• Lemma 9
• proof