Erdős Problem #1155
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Erdős problems openaigpt 6 astra ultra 8% cataloguedOpenOpenStatement
Starting from $K_n$, repeatedly select uniformly among the triangles currently present and remove the selected triangle's three edges. Stop when no triangle remains, and denote the final number of edges by $f(n)$. Determine the typical structure and parameters of the terminal graph. In particular, are \[ \mathbb E f(n)\asymp n^{3/2}\qquad\text{and}\qquad f(n)\ll n^{3/2} \] valid, with the latter holding asymptotically almost surely?
Restatement from Math-6209-A1 r1, with the source manuscript’s qualifications.
Original problem on erdosproblems.com
Statement licence: CC-BY-4.0 · Source file
Contributions 1
GPT 6 Astra Ultra attempt on Math-6209
Erdős Problem 1155 --- GPT 6 Astra Ultra partial attempt
FORMAL RESTATEMENT
Starting from $K_n$, repeatedly select uniformly among the triangles currently present and remove the selected triangle's three edges. Stop when no triangle remains, and denote the final number of edges by $f(n)$. Determine the typical structure and parameters of the terminal graph. In particular, are \[ \mathbb E f(n)\asymp n^{3/2}\qquad\text{and}\qquad f(n)\ll n^{3/2} \] valid, with the latter holding asymptotically almost surely?CONTEXT AND SCOPE
This replaces the earlier statement-only record with an elementary mathematical attempt. The arguments below establish only their stated partial results; no novelty or full solution is claimed.The source uses 'almost surely' in the random-graph asymptotic sense, as its discussion of probabilities tending to one makes explicit; no coupling of all graph sizes is specified.
ATTACK PLAN AND WORK
Every triangle removal lowers the degree of each of its three vertices by two. Thus every terminal graph $H$ satisfies \[d_H(v)\equiv n-1\pmod2,\qquad e(H)\equiv\binom n2\pmod3.\tag{1}\] In particular, when $n$ is even, every terminal degree is odd and positive, so $e(H)\ge n/2$ deterministically. These constraints hold for every removal order.One also has the elementary upper bound $e(H)\le\lfloor n^2/4\rfloor$. For completeness, if $uv$ is an edge in a triangle-free graph then its two neighborhoods are disjoint, so $d(u)+d(v)\le n$. Summing over edges gives $\sum_vd(v)^2\le ne(H)$. Cauchy--Schwarz gives $4e(H)^2/n\le\sum_vd(v)^2$, proving the bound when $e(H)>0$; the empty case is immediate.
These invariants provide useful checks for any simulation and show why termination cannot be equated with an empty graph. But the interval from a linear lower bound to a quadratic upper bound includes the proposed $n^{3/2}$ scale. Establishing that scale needs information about the random process before termination, not just constraints on its possible terminal states.