History of k-core thresholds of the Erdős–Rényi random graph

who	what
2026-09-07 01:45	bmatschke	tagged "statistical mechanics": six tables on critical phenomena carried only "physics" and "combinatorics", which does not distinguish them from anything	current reviewed
2026-09-06 13:33	bmatschke	the arXiv number of 1 references was in the sentence, where it is text; moved to the `arxiv` field, which the page renders as a link to the abstract
2026-09-06 08:03	zeta3	after the critique: "of order n" (which reads as all n vertices) is now "on a positive fraction of the vertices" in the definition and in the history comment; Janson's coauthor is Luczak, no stroke; the 2-core quotient "tends to 1 as lambda -> 0 and increases"; "the same minimisation" names what it
2026-09-06 07:50	zeta3	definition shortened to the object and the formula, with G(n,m) left to the comment on the models; "above c_k" reworded to "for c > c_k", since the audit refuses positional words and this one read as one
2026-09-06 07:47	zeta3 with Claude Code, table-build@bc7	k-core thresholds c_k of the random graph G(n, c/n) for 3 <= k <= 12, the minimum of x / P(Poisson(x) >= k-1) enclosed in ball arithmetic at 100 digits from a bisection bracket around its unique minimiser
2026-09-06 07:47	zeta3	checking that this table can be written to
2026-09-06 07:46	zeta3	draft: k-core thresholds of the random graph, proposal 6 of the 2026-09-06 batch, prose first

from line 123 (63 lines, 60 more than before) @@ -123,3 +123,63 @@

 Display properties:   number-header: $c_k$-Numbers: []+Numbers:+- params:+    k: '3'+  number: '3.350918871511672773156814404987098076190626590909356005328111228070177491045217990747563631554521917'+  comment: The minimum is attained at $\lambda_{3}=1.79328213290$, and the $3$-core,+    when it first appears, has about $0.267580654988\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{3})\geq 3)$ CITE{PSW}.+- params:+    k: '4'+  number: '5.149402746986453309205780306874702219721431580445989663938902883996203674228102712799074136530994881'+  comment: The minimum is attained at $\lambda_{4}=3.38363428285$, and the $4$-core,+    when it first appears, has about $0.438061713058\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{4})\geq 4)$ CITE{PSW}.+- params:+    k: '5'+  number: '6.799275488618085713554858948433881689611001560114832621433538808240774082265154295363651249651653632'+  comment: The minimum is attained at $\lambda_{5}=4.88127749135$, and the $5$-core,+    when it first appears, has about $0.538433561728\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{5})\geq 5)$ CITE{PSW}.+- params:+    k: '6'+  number: '8.365340770047702714643014509591141846278719137911370579394156101303747243609249140526472559924021746'+  comment: The minimum is attained at $\lambda_{6}=6.32250555103$, and the $6$-core,+    when it first appears, has about $0.604638182695\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{6})\geq 6)$ CITE{PSW}.+- params:+    k: '7'+  number: '9.875290724843900398166931013481935649636727369861392447711159365654956909177306398399832460695926528'+  comment: The minimum is attained at $\lambda_{7}=7.72458360044$, and the $7$-core,+    when it first appears, has about $0.651844404355\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{7})\geq 7)$ CITE{PSW}.+- params:+    k: '8'+  number: '11.34412889749009164633003354367252165775865248752217016298709432771144857325090293197553885467849478'+  comment: The minimum is attained at $\lambda_{8}=9.09734440258$, and the $8$-core,+    when it first appears, has about $0.687379687246\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{8})\geq 8)$ CITE{PSW}.+- params:+    k: '9'+  number: '12.78109969599954770002045814494717548620686258246512732253447340375314869612112335175524027145625145'+  comment: The minimum is attained at $\lambda_{9}=10.4470306813$, and the $9$-core,+    when it first appears, has about $0.715208555100\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{9})\geq 9)$ CITE{PSW}.+- params:+    k: '10'+  number: '14.19238948538860551267157564263474164395196391020752665260015321888975561150902701183531116626376871'+  comment: The minimum is attained at $\lambda_{10}=11.7779066065$, and the $10$-core,+    when it first appears, has about $0.737666502714\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{10})\geq 10)$ CITE{PSW}.+- params:+    k: '11'+  number: '15.58238546277987394793632721893316729376409298688945183299417284087190893736485161005949680380499953'+  comment: The minimum is attained at $\lambda_{11}=13.0930424983$, and the $11$-core,+    when it first appears, has about $0.756221714358\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{11})\geq 11)$ CITE{PSW}.+- params:+    k: '12'+  number: '16.95433608082373067537145274656211915227107450158198530770797599423654739794564896542024685566994089'+  comment: The minimum is attained at $\lambda_{12}=14.3947389019$, and the $12$-core,+    when it first appears, has about $0.771845397666\,n$ vertices, the fraction being+    $\mathbb{P}(\mathrm{Po}(\lambda_{12})\geq 12)$ CITE{PSW}.

History of k-core thresholds of the Erdős–Rényi random graph

What changed between 2026-09-06 07:47 and 2026-09-06 07:47