MathDB

Triangular threshold conjecture for simplex layers in the partition graph

Less than 1 year old · traced to

Let GnG_n be the partition graph, let KnK_n be its clique complex, let Lr(n)L_r(n) denote the vertices of local simplex dimension rr, let Δ(n)\Delta(n) be the maximal local simplex dimension, and let τ(r)\tau(r) be the least nn for which the layer Lr(n)L_r(n) is nonempty. Triangular threshold conjecture. For every r≥0r\ge 0,

τ(r)=1+r(r+1)2.\tau(r)=1+\frac{r(r+1)}{2}.

Equivalently,

Δ(n)=max⁡{r: 1+r(r+1)2≤n}.\Delta(n)=\max\Bigl\{r:\ 1+\frac{r(r+1)}{2}\le n\Bigr\}.

The formula is suggested by exhaustive computation for n≤30n\le 30 and predicts the first appearance of successive simplex layers through triangular-number thresholds; its validity for all rr remains open.

References

Primary source

Fedor B. Lyudogovskiy, “Simplex Stratification and Phase Boundaries in the Partition Graph”, arXiv:2603.23228 (2026).

graph-theorypartition-graphssimplicial-complexes
Posted

Progress summary

Refreshed
Claimed solved

A reader-posted argument claims a complete proof of the conjecture, but no independent verification has been found, so the result remains unconfirmed.

Fedor B. Lyudogovskiy posed the conjecture in March 2026: the first occurrence of layer rr should be at τ(r)=1+r(r+1)/2\tau(r)=1+r(r+1)/2, equivalently determining the maximal local simplex dimension at each size.

Known results

  • Exhaustive computation through n≤30n\le 30 gives τ(r)=1,2,4,7,11,16,22,29\tau(r)=1,2,4,7,11,16,22,29 for 0≤r≤70\le r\le 7.
  • For 2≤r≤72\le r\le 7, the first layer consists of one-cell extensions of the staircase partition δr=(r,r−1,…,1)\delta_r=(r,r-1,\ldots,1).
  • Local star and top capacities give an exact characterization of dim⁡loc(λ)\dim_{\mathrm{loc}}(\lambda), but not the global threshold formula.
  • A related synthesis establishes dim⁡loc(δt)=t−1\dim_{\mathrm{loc}}(\delta_t)=t-1 and treats the global pattern as conjectural.

Posted attempt

A reader-posted argument claims a complete proof using explicit corner-incidence formulas and a weight identity for partitions; it claims both the threshold formula and the equivalent maximal-dimension formula. The attempt has not been independently verified.

Current status (as of August 2026): Computation confirms the conjecture through n≤30n\le 30, while a complete-proof claim is unverified and the all-rr statement remains unresolved.

Sources

Solutions 1

ProofThis solution needs a summarySee full solutionHide full solution

Proof of all triangular thresholds and the maximal-dimension formula

The previously established local corner formulas in arXiv:2603.18696, Corollary 3.4, Proposition 3.7, and Theorem 7.1, are used as known input.

Write

λ=(d1m1,…,dkmk),d1>⋯>dk>0,gi=di−di+1,dk+1=0.\lambda=(d_1^{m_1},\ldots,d_k^{m_k}), \qquad d_1>\cdots>d_k>0, \qquad g_i=d_i-d_{i+1}, \qquad d_{k+1}=0.

For removable corner ii and addable corner jj, the known corner-incidence matrix is

Mij=1−1{j=i, mi=1}−1{j=i+1, gi=1}.M_{ij} = 1-\mathbf1_{\{j=i,\ m_i=1\}} -\mathbf1_{\{j=i+1,\ g_i=1\}}.

Hence the star capacities are

si=k+1−1{mi=1}−1{gi=1},s_i=k+1-\mathbf1_{\{m_i=1\}}-\mathbf1_{\{g_i=1\}},

and every top capacity is at most kk. Consequently

dim⁡loc(λ)={0,λ=(1),k−1,λ=(k,k−1,…,1), k≥2,k+1,mi≥2 and gi≥2 for some i,k,otherwise.(1)\dim_{\mathrm{loc}}(\lambda)= \begin{cases} 0,&\lambda=(1),\\ k-1,&\lambda=(k,k-1,\ldots,1),\ k\ge2,\\ k+1,&m_i\ge2\text{ and }g_i\ge2\text{ for some }i,\\ k,&\text{otherwise}. \end{cases} \tag{1}

Put ai=mi−1a_i=m_i-1, bi=gi−1b_i=g_i-1, and Tk=k(k+1)/2T_k=k(k+1)/2. Direct summation gives the exact weight identity

∣λ∣=Tk+∑i=1k(k−i+1)ai+∑i=1kibi+∑1≤i≤j≤kaibj.(2)|\lambda| = T_k+ \sum_{i=1}^k(k-i+1)a_i + \sum_{i=1}^k i b_i + \sum_{1\le i\le j\le k}a_i b_j. \tag{2}

All terms after TkT_k are nonnegative.

Fix r≥2r\ge2. If k≤r−2k\le r-2, (1) rules out local dimension at least rr. If k≥r+1k\ge r+1, (2) gives

∣λ∣≥Tr+1>Tr+1.|\lambda|\ge T_{r+1}>T_r+1.

If k=rk=r, the staircase itself has dimension r−1r-1, so dimension at least rr requires

∣λ∣≥Tr+1.|\lambda|\ge T_r+1.

Equality occurs only for b1=1b_1=1 or ar=1a_r=1, with all other excess parameters zero.

If k=r−1k=r-1, dimension at least rr requires ai,bi≥1a_i,b_i\ge1 for a common ii. Formula (2) then gives

∣λ∣−Tk≥(k−i+1)+i+1=k+2,|\lambda|-T_k \ge(k-i+1)+i+1=k+2,

hence again

∣λ∣≥Tk+k+2=Tr+1.|\lambda|\ge T_k+k+2=T_r+1.

Equality occurs exactly when ai=bi=1a_i=b_i=1 for one 1≤i≤k1\le i\le k, with all other excess parameters zero.

Both equality constructions exist. Therefore

τ(r)=r(r+1)2+1(r≥2).\boxed{\tau(r)=\frac{r(r+1)}2+1\qquad(r\ge2).}

The exceptional initial layers satisfy τ(0)=1\tau(0)=1 and τ(1)=2\tau(1)=2, as required.

Moreover, for every n≥Tr+1n\ge T_r+1, the partition

(n−Tr+r,r−1,r−2,…,1)(n-T_r+r,r-1,r-2,\ldots,1)

has exactly rr distinct part sizes and local dimension rr. Conversely, the argument above rules out dimension rr at every smaller size. Hence

Δ(n)=max⁡{r≥0:r(r+1)2+1≤n}.\boxed{ \Delta(n)=\max\left\{ r\ge0:\frac{r(r+1)}2+1\le n \right\}. }

This proves every assertion of the triangular-threshold conjecture.

0
Shivam Patel Posted