621.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 621

References:

erdosproblems.com/621
[EGT96] Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, Covering and independence in triangle structures. Discrete Math. (1996), 89-101.
[Er99] Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6.
[NoSu16] S. Norin and Y.-R. Sun, Triangle-free independent sets vs. cuts. arXiv:1602.04370 (2016).

Imports (1)

FormalConjecturesUtil

theorem Erdos621.erdos_621 Documented declaration: erdos_621 sorry line 64 source
Source documentation
Let $G$ be a graph on $n$ vertices, $\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle.
Is it true that$$\alpha_1(G)+\tau_1(G) \leq \frac{n^2}{4}?$$
A problem of Erdős, Gallai, and Tuza [EGT96], who observe that this is probably quite difficult since there are different examples where equality hold: the complete graph, the complete bipartite graph, and the graph obtained from $K_{m,m}$ by adding one vertex joined to every other.
This is true, and was proved by Norin and Sun [NoSu16], who in fact proved that$$\alpha_1(G)+\tau_B(G) \leq \frac{n^2}{4},$$where $\tau_B(G)$ is the minimum number of edges that need to be removed to make the graph bipartite.
Here $\alpha_1(G)$ and $\tau_1(G)$ are taken over subsets of the edge set of $G$, and the inequality is stated multiplied through by $4$ so that it lives in the natural numbers.
The linked file states $\tau_1$ as the least number of edges whose deletion leaves $G$ triangle-free, which is the same as meeting every triangle of $G$, and quantifies over an arbitrary Fintype V rather than Fin n.

depends on sorry what this means
Statement uses: Iff, True, Nat, SimpleGraph, Fin, IsGreatest, Set.ofPred, Exists, Finset, Sym2, And, LE.le, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, Eq, Finset.card, Inter.inter, Insert.insert, Sym2.mk and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_621 : answer(True) ↔
    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (a t : ℕ),
      IsGreatest {k : ℕ | ∃ A ⊆ G.edgeFinset, A.card = k ∧
        ∀ x y z : Fin n, G.Adj x y → G.Adj y z → G.Adj x z →
          (({s(x, y), s(y, z), s(x, z)} : Finset (Sym2 (Fin n))) ∩ A).card ≤ 1} a →
      IsLeast {k : ℕ | ∃ T ⊆ G.edgeFinset, T.card = k ∧
        ∀ x y z : Fin n, G.Adj x y → G.Adj y z → G.Adj x z →
          (T ∩ ({s(x, y), s(y, z), s(x, z)} : Finset (Sym2 (Fin n)))).Nonempty} t →
      4 * (a + t) ≤ n ^ 2 := by
…
```